High Resolution Observations and Modeling of
MG0414+0534
by
John D. Ellithorpe
B.S. Physics, University of California, Irvine (1990)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1995
() Massachusetts Institute of Technology 1995. All rights reserved.
--
Author
Department of Physics
May 18, 1995
Certified by.
I-
I
d
v
Jacqueline N. Hewitt
Class of 1948 Associate Professor of Physics
Thesis Supervisor
Accepted by
George F. Koster
nOF
TCI4t4L0GY LTEuChairman,
JUN 2 61995
LIBRARIES
Graduate
Committee
High Resolution Observations and Modeling of
MG0414+0534
by
John D. Ellithorpe
Submitted to the Department of Physics
on May 18, 1995, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
Gravitational lenses provide unique opportunities to probe distant galaxies and to
examine models of the universe. We focus our attention on the gravitational lens
MG 0414+0534. The bright four-image geometry and source variability make this
system an excellent candidate for study. The first step in addressing astrophysical
applications is the accurate determination of the lens matter distribution. In this
thesis, we focused on improving the lens inversion algorithms and obtaining highresolution observations of MG 0414+0534 to yield a reliable reconstruction of the
lens. We then use this information to measure astrophysical properties of the lens
and universe.
Multiple imaging by the lens provides strong constraints in the observed images
allowingreconstruction of both the total matter distribution in the lens and the light
distribution in the source. Although the lens inversion algorithm, LensClean, has
been applied successfully, we find systematic errors which bias the results. Systems
with compact images are affected to a greater extent, though the effects are also significant in systems dominated by diffuse emission. The primary errors result from the
inclusion of negative clean components and the assumption that the standard image
reconstruction procedures do not affect the lens modeling. We develop the Visibility
LensClean algorithm which removes the systematic errors by requiring positive clean
components and by operating directly on the complex visibility data. We find the Visibility LensClean algorithm dramatically reduces the systematic errors and provides
more reliable lens inversion results. This algorithm also yields a better reconstruction
than the standard image reconstruction techniques once a lens model is found which
fits the system reasonably well.
Using 15 GHz (A2cm) VLA observations, we find that a simple monopole plus
quadrupole structure is insufficient in describing the lens in MG 0414+0534. The
angular structure of the lens models does not produce the observed image geometry,
0'1. Since the nature of LensClean requires that
resulting in a position error of
the nonlinear equation for the images be solved efficiently, we can not explore a more
general class of lens models within the LensClean framework at present. Instead,
using a point source inversion procedure, we find the lens is fit satisfactorily by a
singular isothermal sphere plus the two lowest order terms in the multipole expansion due to an mass distribution outside the ring of images, 17r2 cos 2(0 - ,) and
/Or3 cos 3(0 - 0).
We observed MG 0414+0534 at 5 GHz (A6cm) using Very Long
Baseline Interferometry and detected all the components except C. We find resolved
structures in Al, A2, and B at the 10 milliarcsecond level. We modeled the lens using
the two strongest components in each VLBI image and find the cos 30 model does fit
the VLBI data and the best fit lens parameters are consistent with those found with
the 15 GHz VLA data.
We believe the galaxy detected by Schechter & Moore (1993) is the primary object
responsible for the lensing. The orientation of the quadrupole moment is consistent
with the shear caused by other objects present in the field, although the inferred
masses of those objects would be much larger than expected for normal galaxies. We
believe the cos 30 perturbation to the ellipticity is due to object X. If we assume an
Einstein-de Sitter cosmology and a lens redshift of ZL = 0.5, we find the mass of the
1 M and the velocity dispersion is 258 + 1 km/s.
primary galaxy is 8.5 + 0.1 x 10 1h75
These values are consistent with typical elliptical galaxies. The best fit lens model
predicts 0.6
0.8 for the A2/A1 magnification ratios of the VLBI components. If
the quasar core is associated with these components, the predicted magnification
ratios cannot explain the optical A2/A1 flux ratio of 0.3. We estimate the apparent
magnitude of the galaxy and find that if the lens is a normal elliptical galaxy then it
contains a significant amount of dust or is at a large redshift (ZL
1).
Assuming a time delay of 2 weeks for the A1-B time delay and lens redshift of
ZL = 0.5, the best fit lens model predicts Ho = 60 ± 8 km s - 1 Mpc-' in an Einsteinde Sitter cosmology. The exact value of Ho is model dependent, and constraining
the radial shape of the lens is necessary to provide a unique relationship between the
time delay and the Hubble constant. If Ho is measured independently, this system
may be able to put limits on Q and possibly A. Deep high-resolution observations at
optical and radio wavelengths should provide more stringent constraints on the lens
inversion and may reveal more information about the nature of object X. Furthermore,
performing the lens inversion with the Visibility LensClean algorithm will reduce the
possible range of lens models by using all the available information in the observations.
Thesis Supervisor: Jacqueline N. Hewitt
Title: Class of 1948 Associate Professor of Physics
To the memory of my father
Acknowledgments
I wish to thank those people who have helped me during my tenure at the Massachusetts Institute of Technology. I thank my advisor, Professor Jacqueline Hewitt,
for allowing me to pursue the research in this thesis. She has embarked on many
intriguing research topics during the course of my thesis and I am sincerely grateful
to have worked with her. I also thank Professor Edmund Bertschinger and Professor
Peter Fisher for participating on my thesis committee. Professor Bertschinger gave
his time and provided many useful suggestions. Even though Professor Fisher is not
an astrophysicist, his perspective and suggestions improved the quality of this thesis. I must also thank Professor Kerson Huang for being on my thesis committee for
over a year. I regret that Professor Huang was not able complete his service on my
committee due to his sabbatical this semester.
I thank Professor Christopher Kochanek at the Harvard-Smithsonian Center for
Astrophysics. I am very grateful for his many useful discussions and scientific insights
which helped me over the past two years. I am privileged to have been able to work
with him.
I wish to thank all those who have worked in the radio astronomy group at MIT
over the past five years. I have enjoyed working with Grace Chen, Charlie Katz, and
Christopher Moore. My years at MIT would not have been the same without Max
Avruch, Sam Conner, and with the rest of Professor Burke's group. I would also like
to thank Mike Titus at the Haystack Observatory. His assistance and friendship was
invaluable during the correlation of the 5 GHz VLBI observation.
I thank Krishna Shenoy for his many years of friendship and support.
I have
enjoyed our many discussions and adventures both at UC Irvine and MIT. I also
thank my family for their years of love and support. I especially thank my father for
his continuous belief in my abilities. Finally, and most importantly, I want to thank
Elizabeth Barton for her love and understanding and Elsie for her constant attention.
Contents
Abstract
2
Acknowledgments
5
1
Introduction
20
1.1 The MG 0414+0534 System .......................
21
1.2 Motivation.
................................
24
............................
26
1.3
Chapter Summary
2 Interferometry Theory
27
................... 27
2.1
Introduction.
2.2
The Two-Element Interferometer
. . . ..
2.3
Aperture Synthesis.
. . . . . . . . . . . . . . . . . . . .
2.4
Smearing Effects.
. . . . . . . ..
2.4.1
Bandwidth Smearing .
. . . . . . . . . . . . . . . . . . . .
34
2.4.2
Time-Average Smearing
. . . . . . . . . . . . . . . . . . . .
35
2.5
2.6
..
Image Reconstruction .
. . . ..
..
2.5.1
Self-Calibration ......
. ..
..
.......
2.5.2
Deconvolution Algorithms . ..
..
..
Very Long Baseline Interferometry
..
..
. . . . . . . . . . . TT28
. ..
. ..
. . ..
. . TT34
. . . . . . . . TT36
..
..
..
. . . ..
..
...
. TT37
. . . . . . TT38
. . . . . . . . . . . . . . . . . . .
41
41
2.6.1
Amplitude Calibration
. . . . . . . . . . . . . . . . . . . .
2.6.2
Fringe Fitting.
. . . . . . . . . . . . . . . . . . . .
6
31
T42
3 Gravitational Lensing and Lens Inversion
3.6.1
MG 0414+0534 ..............
........... ..46
........... . 57
........... ..59
........... ..63
........... ..66
........... . 67
3.6.2
MG 1654+1346 ..............
..
3.1
Gravitational Lens Theory ............
3.2
Lens Inversion ...................
3.3
Clean map LensClean.
3.4
Negative Clean Components ...........
3.5
Visibility LensClean.
3.5.1
3.6
3.7
..
..............
Clean map LensClean vs Visibility LensClean
Discussion of LensClean .............
Visibility LensClean.
4.1.1
Lens Models ......
4.1.2 Results .........
4.2
Point Source Lens Inversion .
4.2.1
Lens Models ......
4.2.2 Results .........
5 5 GHz VLBI Observations
5.1
Observations ..........
5.2
Correlation
..........
5.3 Calibration ..........
5.3.1 Fringe Fitting .....
5.3.2 Amplitude Calibration
5.4
..
..
.....
. .54
Self-Calibration with a Model Consistent with Lensing ....
4 Modeling the 15 GHz VLA Dat
4.1
46
Image Reconstruction .....
5.4.1 Editing .........
5.4.2
Mapping Procedure . .
5.4.3
0423-013 ........
5.4.4
MG 0414+0534 ....
..
..
.....
66
. .71
.. .. .... ... . .78
81
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
7
81
82
83
91
94
95
102
102
103
105
105
106
107
107
110
112
112
5.5
5.6
6
Image Reliability.
116
5.5.1
Consistency within the Data Sets ....
117
5.5.2
Time Segmentation.
117
5.5.3
Consistency with Gravitational Lensing.
119
Lens Inversion ...................
122
Conclusions
128
6.1
Lens Inversion Results.
6.2
Optical-Radio Flux Ratio Discrepancy
6.3
Nature of the Lens ................
133
6.4
Prospects for Determining H .
138
6.5
Future Prospects
128
.....
132
.................
140
A Lens Models
142
A.1 Singular Isothermal Sphere (SIS) plus External Shear .
A.2 Generalized Monopole plus External Shear .......
A.3 Approximate Singular Isothermal Ellipsoid .......
A.4 De Vaucouleurs Mass Distribution plus External Shear
A.5 Approximate Elliptical Mass Distribution .
A.6 SIS plus External Shear and Cosine 30 Component
A.7 SIS plus External Shear and Octupole
Bibliography
.........
. .
... . 142
... . 143
....
....
145
146
... . 147
... . 149
....
150
152
8
List of Figures
1-1 Total Intensity Map of MG 0414+0534 from 15 GHz VLA Observations
(Katz & Hewitt 1995). The contour levels are -0.375, 0.375, 0.75, 1.5,
3, 6, 12, 24, 48, and 96 percent of the peak, 160.7 mJy/Beam.
The
x symbol shows the position of the lens galaxy (Falco 1995). The +
symbol shows the position of Object X (Schechter & Moore 1993). . .
22
2-1 Diagram of the two element interferometer. b is the vector from antenna I to antenna 2, s is the direction to the source, rg is the geometric
time delay, and rOis a delay introduced before correlation.
......
29
3-1 Gravitational lens diagram. The path of a single light ray is drawn.
Dij is the angular diameter distances from point i to point j, 0 is
the angular position of the image (apparent source),
/
is the angular
position of the source in the absence of a lens, and at is the deflection
angle
....................................
47
3-2 The deflection angle diagram for a point lens of mass M. The axes are
in units of the "Einstein" ring radius ...................
49
3-3 The deflection angle diagram for a general lens with circular symmetry. 50
3-4 The caustics and critical lines for a strong non-singular elliptical lens.
Panel (a) shows the source plane caustics. Panel (b) shows the image
plane critical lines. The c, , and x symbols represent sources in the
one, three, and five image regions, respectively ..............
9
53
3-5 Final CLEAN image of MG 0414+0534 from a 15 GHz VLA observation of MG 0414+0534 (Katz & Hewitt 1995) used in the Clean map
LensClean analyses. The contour levels are -1.5, -0.75, -0.375, 0.375,
0.75, 1.5, 3, 6, 12, 24, 48 and 96 percent of the peak, 158.4 mJy/Beam.
The beam FWHM is 0'.12 by 0'l11. ...................
59
3-6 Residual Map from Best Fit SIS plus external shear model using the
Clean map LensClean method including negative clean components.
The absolute contour levels are the same as Figure 3-5 .........
60
3-7 Distribution of clean components used in the best fit SIS plus external
shear model using the Clean map LensClean method including negative
clean components ..............................
62
3-8 Multiplicity image from the best fit SIS plus external shear model
using the Clean map LensClean method including negative clean components. The curves are boundaries between the 1, 2, and 4 image
regions. Al, A2, B, and C show the locations of the MG 0414+0534
components and + symbol shows the location of the clean components
west of B. ................................
63
3-9 The MG+0414+0534 residual images after LensClean. The absolute
contour levels are the same as Figure 3-5. Panel (a) shows the best
fit residuals after Clean map LensClean. Panel (b) shows the residuals
after Visibility LensClean using the best fit Clean map LensClean lens
model. Panel (c) shows the best fit residuals after Visibility LensClean.
Panel (d) shows the best fit residuals after Visibility LensClean on the
self-calibrated data set ...........................
68
3-10 Final CLEAN image of MG 1654+1346 used in Clean map LensClean.
The contour levels are -3, -1.5, 1.5, 3, 6, 12, 24, 48, and 96 percent of
the peak, 5.01 mJy/Beam. The beam FWHM is 0'.'21 by 0'.'19. ....
10
72
3-11 The MG 1654+1346 residual images after LensClean. The absolute
contour levels are the same as Figure 3-10. Panel (a) shows the best
fit residuals after Clean map LensClean. Panel (b) shows the best
fit residuals after Visibility LensClean. Panel (c) shows the best fit
residuals after Visibility LensClean on the self-calibrated data set. ..
74
3-12 The distribution of clean components using the Clean map LensClean
algorithm on the 5 GHz MG 1654+1346 data set. Panel (a) shows the
distribution using the constraint of positive definite clean components
and Panel (b) shows the distribution with negative clean components.
The + symbols show the positions of the largest negative clean components (
15 ptJy/Beam) .........................
76
3-13 The reconstructed Visibility LensClean images of MG 1654+1346 using
the self-calibrated data set. The absolute contour levels are the same as
Figure 3-10. Panel (a) shows the reconstruction at normal resolution,
0'21 by 019. Panel (b) shows the super-resolved reconstruction using
a 01 restoring beam ............................
78
4-1 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit singular isothermal sphere (SIS) plus external shear model (1). The absolute contours levels are the same as
Figure
3-5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4-2 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit approximate singular isothermal ellipsoid
model (3). The absolute contours levels are the same as Figure 3-5.
4-3
86
Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit generalized monopole plus external shear
model (2) using a = 0.0 and s = 0''005. The absolute contours levels
are the same as Figure 3-5 .............
11
............
86
4-4 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit generalized monopole plus external shear
model (2) using a = 0.5 and s = 0'.'005. The absolute contours levels
are the same as Figure 3-5 ........................
87
4-5 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit generalized monopole plus external shear
model (2) using a = 1.0 and s = 0'.'005. The absolute contours levels
87
are the same as Figure 3-5 ........................
4-6 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit generalized monopole plus external shear
model (2) using a = 1.0 and s = 0'.'05. The absolute contours levels
88
are the same as Figure 3-5 .....................
4-7 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit generalized monopole plus external shear
model (2) using a = 1.0 and s = 0'.'20. The absolute contours levels
are the same as Figure 3-5 .....................
88
4-8 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit de Vaucouleurs plus external shear model
(4) using Re = 1'.'25. The absolute contours levels are the same as
Figure 3-5.
.
. . . . . . . . . . . . . . . . . . . . . . .
....
89
4-9 Panel (a) shows the reconstructed image and Panel (b) shows the residual image from the best fit approximate elliptical mass distribution
model (5) using a = 0.5. The absolute contours levels are the same as
Figure
3-5. . . . . . . . . . . . . . . . . .............
89
4-10 Panel (a) shows the reconstructed source and tangential caustic and
and Panel (b) shows the point images and the tangential critical for
Model 4. The image in panel (a) is a factor of four smaller in each
dimension than the image in panel (b). Due to the finite pixel size, the
source components do not lie on top of one another ..........
12
97
4-11 Panel (a) shows the reconstructed source and tangential caustic and
and Panel (b) shows the point images and the tangential critical for
Model 8. The image in panel (a) is a factor of four smaller in each
dimension than the image in panel (b). Due to the finite pixel size, the
source components do not lie on top of each other ............
100
4-12 Plot of X2 versus the flux of component B. We used the cos30 lens
model. The X2 is the value after optimizing all the lens model parameters. The D symbol indicates the measured flux of component B used
in the lens inversion. ...........................
101
5-1 Visibility amplitude for 0423-013 (J2000) on Haystack baselines. The
amplitudes dramatically drop after 13:00 UT.
.............
109
5-2 The left panel shows the u-v coverage and the right panel shows the
dirty beam for the 5 GHz (A6cm) observation of MG 0414+0534 after
the data were edited. To produce the dirty beam, the data were uniformly weighted with a u-v box size of 3 pixels. The FWHM of the
central lobe is 17 mas by 8 mas. The largest sidelobe is 50 percent of
the peak. . . . . . . . . . . . . . . .
.
. . . . . . . . . ......
110
5-3 Final 5 GHz image of 0423-013 (J2000). The data have been uniformly
weighted with a u-v box size of 3 pixels. The size of the restoring beam
is given in the lower left corner. ......................
113
5-4 Final 5 GHz image of each component in MG 0414+0534. Clockwise
from the bottom left, the images are Al, A2, B, and C. The data have
been uniformly weighted with a u-v box size of 3 pixels. The contour
levels are -2, -1, 1, 2, 4, 8, 16, 32, 64, and 96 percent of the peak flux
in Al (185 mJy/Beam).
The size of the restoring beam is shown in
the lower left corner of each panel. The image parameters are given in
Table
5.3.
. . . . . . . . . . . . . . . .
13
.
. . . . . . . . ......
115
5-5 Final 5 GHz images of A1/A2 and B using data from 16:00 UT to
23:00 UT. The data have been uniformly weighted with a u-v box size
of 3 pixels. The contour levels are -3, -1.5, 1.5, 3, 6, 12, 24, 48, and
96 percent of the peak in Al (193.8 mJy/Beam).
The size of the
restoring beam is shown in the lower left corner of each panel
....
118
5-6 The upper right panel shows the reconstructed source brightness distribution of MG0414 from the clean component model of Al.
The
reconstructed source has been convolved with the lens distorted clean
beam. The other panels contain the reconstructed images of MG0414
using the reconstructed MG0414 source. The source and images were
created using the Gravitational Image Tracing (GrIT) package. The
brightest two components are indicated in each panel with the Roman
numeral: I is the brightest component and II is the second brightest
component
at Al.
. . . . . . . . . . . . . . . .
.
..........
121
5-7 The reconstructed images for the best fit cos30 model. The upper
right panel shows the reconstructed source brightness distribution of
MG0414 from the clean component model of Al. The reconstructed
source has been convolved with the lens distorted clean beam. The
other panels contain the reconstructed images of MG0414 using the
reconstructed MG0414 source. The source and images were created
using the Gravitational Image Tracing (GrIT) package. The brightest
two components are indicated in each panel with the Roman numeral:
I is the brightest component and II is the second brightest component
at Al. ....................................
126
6-1 Mass of the primary lensing galaxy as a function of lens redshift. We assume a Friedmann cosmology with A = 0 and Ho = 75 km s-l Mpc-l.
We plot three curves corresponding to Q = 0.2, 1.0, and 1.5. .....
134
6-2 Velocity dispersion of the lensing galaxy as a function of lens redshift
and Q. We plotted three values of Q: 0.2, 1.0, and 1.5.
14
........
136
6-3 Apparent magnitude of the lensing galaxy as a function of lens redshift
and Q using the Faber-Jackson relation which assumes the lens galaxy
is a normal elliptical galaxy. We have assumed Ho = 75 km s- 1 Mpc -1
and plotted curves three values of Q: 0.2, 1.0, and 1.5. ........
137
6-4 Hubble constant for different lens models. We have assumed a time
delay of 2 weeks between Al and B. We have assume
= 1 and A = 0. 139
6-5 Hubble constant for different cosmologies and best fit lens model. We
have assumed a time delay of 2 weeks between Al and B ........
15
140
List of Tables
3.1
Results for the MG 0414+0534 system using the singular isothermal
sphere plus external shear model. VLC/SC indicates the results using
the visibility data set after self-calibration with the converged VLC
model. The lens position is measured relative to component B. P-P
indicates the peak-to-peak value in the residual image. 0y is measured
north through east. The CLC confidence intervals are calculated using
AX2 < 15.1 and the VLC confidence intervals are calculated using
AX2 < 4. X2 is calculated using Xc2 for the CLC model and X21, for
the VLC models. .............................
69
3.2 Results for the MG 1654+1346 system using the approximate elliptical
mass distribution.
VLC/SC indicates the results using the visibility
data set after self-calibration with the converged VLC model. The
lens position is measured relative to the quasar core. P-P indicates the
peak-to-peak value in the residual image. 0y is measured north through
east. The CLC confidence intervals are calculated using AX2 < 15.1
and the VLC confidence intervals are calculated using AX2 < 4. X2 is
calculated using Xc.C for the CLC model and XvCfor the VLC models.
16
75
4.1
Visibility LensClean results on the 15 GHz VLA MG 0414+0534 data
set. The results are from Visibility LensClean on the original selfcalibrated data set. The errors are 95 percent confidence levels. The
lens position is measured relative to component B. The shear position
angle is measured north through east. P-P is the peak-to-peak value
in the residual image ............................
4.2
84
Image positions and peak fluxes for the components in MG0414. The
image positions are measured relative to component B. These data
were taken from Katz & Hewitt (1995) ..................
4.3
93
Best fit lens model parameters and error statistics for the singular
isothermal sphere plus external shear and approximate singular isothermal ellipsoid using the point source solver. The errors are 95 percent
confidence intervals. The lens position is measured relative to component B. These models have six degrees of freedom ............
4.4
95
Best fit lens model parameters and error statistics for lens models 3,
4, and 5 using the point source solver. The errors are 95 percent
confidence levels. The lens position is measured relative to component
B. These models have four degrees of freedom ..............
4.5
96
Best fit lens model parameters and error statistics for lens models 6,
7, and 8 using the point source solver. The errors are 95 percent confidence levels. The lens positions are measured relative to component
B. Model 6 has five degrees of freedom and models 7 and 8 have three
degrees of freedom. .........................
5.1
99
Correlation delay center coordinates. The MG 0414+0534 positions are
taken from Katz & Hewitt (1995) and the 0423-013 position is from
the VLA Calibrator Manual. All positions measured with respect to
the J2000 coordinate system ........................
17
104
5.2
Four point source model of MG 0414+0534 used for the global fringe
fitting. The positions and flux ratios are relative to B. The absolute
positions and relative fluxes were obtained from Katz & Hewitt (1995)
and the absolute flux level was determined from Hewitt et al. (1992).
All positions are measured with respect to J2000 coordinates.
5.3
Parameters from the 5 GHz VLBI images of MG 0414+0534.
....
107
The
positions (Aat,AS) are measured from the peak in the image to the
peak in the B image. ca is determined in empty regions of the final
images. The 5 GHz VLA integrated fluxes are taken from Katz &
Hewitt (1995) ................................
5.4
116
Parameters from the 5 GHz VLBI images of MG 0414+0534 using only
the data from 16:00 UT to 23:00 UT. a is determined in empty regions
of the final images. t The noise level for Al and A2 was determined
from the same image.
5.5
. . . . . . . . . . . . . . . .
.
........
119
Parameters for the singular isothermal sphere plus external shear lens
model used to ray-trace the Al clean component model.
The lens
position (x,y) is measured relative to B and x increases to the west.
v
is measured north through east ......................
5.6
Input positions and fluxes from the VLBI images. The component
positions are measured with respect to the B primary component. ..
5.7
120
123
Best fit lens model parameters and error statistics for the cos 30 model
on the 5 GHz VLBI data. P means only the primary components were
fit. P&S 1 includes both the primary and secondary components were
used in the inversion and P&S 2 includes the primary component and
only the fluxes of the secondary component. P&S+C is identical to
P&S 1 except that the VLA C position was added as a constraint.
The errors are 95 percent confidence levels. The lens position is measured relative to the primary component in B. The position angles are
measured north through east.
.......................
18
124
5.8
The relative separations between the primary and secondary components for both the 5 GHz VLBI data set and best fit lens model. ....
6.1
125
Parameters for lens model which best fit the MG 0414+0534 system,
the singular isothermal sphere plus external shear and cos 30 component. The errors are 95 percent confidence levels. The lens position is
measured relative to the primary component in B. The position angles
are measured north through east. .....................
6.2
129
Magnifications at the positions of the primary and secondary compo-
nents predicted by the best fit cos 30 lens model.. . . . . . . . ....
19
132
Chapter 1
Introduction
Long before the development of the general theory of relativity, it was speculated
that light rays would be deflected by a massive body in the same manner as normal
particles (Michell 1784; Laplace 1795). Soldner (1804) predicted that if the light from
a star grazed the surface of the sun, the apparent position of the star would move
by 0.85 arcseconds. Einstein (1915) applied his general theory of relativity to this
problem and predicted that, due to the curvature of space and time dilation, a ray of
starlight would be deflected by twice that predicted by the classical methods. During
a solar eclipse in 1919, Eddington measured the apparent shift of stars, thus proving
light is affected by gravity (Eddington 1919; Dyson et al. 1920). The magnitude of
the shifts ruled out the Newtonian description and provided a strong confirmation
of general relativity. Using interferometry, these observations have been refined such
that the agreement with general relativity is better than a tenth of a percent (Lebach
et al. 1995).
Eddington (1920) and Chowlson (1924) soon realized that the gravitational effect
can cause multiple images of a single background source. They showed that if a star
lies sufficiently close to the line-of-sight of another more distant star, a second image
of the background star will appear. In the dramatic case of a star positioned directly
behind an intervening star, the image of the background star will be distorted into
an entire ring (called an "Einstein" ring). However, Einstein (1936) calculated the
image separations would be too small to observe. Zwicky (1937a) was the first to
20
recognize that the lensing of galaxies by galaxies would lead to detectable deflections.
He stated that the probability of finding a gravitational lensing system is "practically
a certainty" because of the masses and distances involved (Zwicky 1937b). He was
also the first to realize the importance of gravitational lenses as a probe of distant
galaxies and cosmology.
Even though Zwicky predicted that gravitational lenses should be easily detected,
the first discovery of a lens system did not occur until 1979. Walsh, Carswell, & Weymann (1979) observed the "twin QSO" system, 0957+561, and found that both QSO
images, separated by 6 arcseconds, had a redshift of z = 1.41 and nearly identical spectra. From this spectroscopic evidence they concluded that it was very unlikely that
the images were of separate objects. Since this discovery, many different gravitational
lens systems have been found: point-like systems, arcs, and rings. The point-like systems consist of two or more compact images of a distant quasar, such as 0957+561
and MG0414+0534 (Hewitt et al. 1988a). The arcs are usually lensed images of faint
blue galaxies by a cluster. The blue color distinguishes the arcs from intrinsic structures within the cluster. Examples are seen in Abell 370 (Lynds & Petrosian 1986)
and Abell 963 (Lavery & Henry 1988). The most dramatic examples of gravitational
lensing are the ring systems (also referred to as "Einstein" rings). The lobe of a distant radio galaxy is lensed into a ring of emission, such as in MG1131+0456 (Hewitt
et al. 1988b) and MG1654+1346 (Langston et al. 1989). The rings are always images
of an extended background object, since the probability of finding a perfect alignment
of a compact object necessary to produce a true "Einstein" ring is effectively zero.
1.1
The MG 0414+0534 System
In contrast to the early discoveries of gravitational lenses, MG 0414+0534 (hereafter
MG0414) was discovered by a systematic search for lenses using high-resolution radio
observations (Hewitt et al. 1988a). MG0414 was selected as a gravitational lens
candidate because of its unusual radio structure, consisting of three bright objects.
'Through high-resolution VLA observations, Hewitt et al. (1992) discovered the system
21
actually comprised four images in a configuration suggestive of lensing of a compact
object by a galaxy. Figure 1-1 shows the arcsecond-scale radio structure of MG0414
B'O
0
O
Object X
.
o
a
Galaxy
o
-
A2~
0
A"
:
1
:
0
-1
Relative R.A. (arcsec)
Maximum 0.1607 JY/BEAM
Contou (): -0.38 0.38 0.75 1.50 3.00 6.00 12.00 24.00 48.00 96.00
Beam:FWHM
0.13 x 0.11 (arcso), p.o. -13.0'
Figure 1-1: Total Intensity Map of MG 0414+0534 from 15 GHz VLA Observations
(Katz & Hewitt 1995). The contour levels are -0.375, 0.375, 0.75, 1.5, 3, 6, 12, 24, 48,
and 96 percent of the peak, 160.7 mJy/Beam. The x symbol shows the position of the
lens galaxy (Falco 1995). The + symbol shows the position of Object X (Schechter
& Moore 1993).
from 15 GHz Very Large Arrayl observations in January 1993 (Katz & Hewitt 1995).
The close source-lens alignment in MG0414 leads to four images of the source, rather
than the two images seen in the "Twin Quasar" gravitational lens system, 0957+561.
All the components except C show extended structure. The Al and A2 components
have low-level emission extending towards each other.
Component B is elongated
along the NW-SE direction, approximately in a tangential direction with respect to
1
The Very Large Array is part of the National Radio Astronomy Observatory which is operated
by Associated Universities, Inc. under co-operative agreement with the National Science Foundation.
22
the lens that is near the center of the system.
If the lens has a smooth surface mass density, the number of images in the system
must be odd (Schneider et al. 1992). The "odd" or "fifth" image is expected to lie near
the lens position. Deep radio observations of MG0414 have not detected any emission
from either the "fifth" image or the lensing galaxy (Katz & Hewitt 1995). The limits
imposed on the "fifth" image magnification are useful since they correspond directly
to upper limits on the central surface mass density (Kochanek 1991).
The flux in MG0414 is dominated by Al and A2, which together contain 78 percent
of the radio flux. Component B is the next brightest with 16 percent, and C has only
6 percent. The gravitational lensing effect is achromatic. In other words, if a source
emits radiation at two frequencies, the deflection and magnification observed at both
frequencies will be the same. A lens candidate can be confirmed by observations
at multiple frequencies. This method assumes the intrinsic source structure does not
vary with frequency and that additional effects such as dust extinction or microlensing
by the lens galaxy are not present. Katz & Hewitt (1995) found the flux ratios at
many radio wavelengths are equal, even though the total flux of the system changes
rapidly with a steep spectral index of a = 0.8 (f
v-a).
Hence, there is little
observational evidence for any absorption or microlensing at radio wavelengths. The
radio flux ratios are expected to approximate the true lensing magnifications because
the typical sizes of the radio sources are large compared with the expected crosssection for microlensing events (Witt et al. 1994; Blandford 1990).
Optical observations of MG0414 detected the same structure as seen in radio
images, confirming MG0414 as a true gravitational lens (Hewitt et al. 1992). The
relative positions of all components in optical images agree with those measured in the
radio (Schechter & Moore 1993). The optical flux ratios, however, are not consistent
with those measured at radio wavelengths. Lawrence et al. (1994) explained the flux
ratio discrepancy through differential amounts of dust in the lensing galaxy. They
found the background source is best fit by a z = 2.64 flat-spectrum quasar seen
through Av = 4.5
6 magnitudes of visual extinction.
However, Witt, Mao, &
Schechter (1994) found that microlensing can explain the optical flux ratios. Optical
23
monitoring of MG0414 is necessary to resolve the competing theories, since we expect
that microlensing events will change over a timescale of months.
Schechter & Moore (1993) detected a single galaxy at the center of MG0414,
which they believed responsible for the lensing. Optical and infrared spectroscopy
have not found any features attributed to the lensing galaxy (Lawrence 1995; Hewitt et al. 1992). To the west of component B, a faint component (called "object
X") has been observed (Schechter & Moore 1993; Angonin-Willaime et al. 1994).
The nature of this object has not been determined although Angonin-Willaime et
al. (1994) suggest that it is a distant galactic star. Observations with the Hubble
Space Telescope show diffuse emission rather than a compact object at the position
of object X (Falco 1995), implying that it is a galaxy. If object X is extragalactic
and contains significant mass, its proximity to the line-of-sight will affect the simple
lensing scenario.
1.2
Motivation
Gravitational lens systems can provide unique information about distant galaxies and
cosmology. The most direct application is the determination of matter distributions
in the lens galaxies. The rotation curves of spiral galaxies and velocity dispersion in
clusters indicate significantly more mass than is inferred from summing the luminous
matter.
The gravitational lens effect is dependent only on the total matter distri-
bution in the lens. Hence, reconstruction of the lens provides a probe into the dark
matter distribution in the lensing galaxy (Kochanek 1995, for example).
Another intriguing use of gravitational lenses is as a cosmological ruler. Consider a lensing system which contains multiple images of a common source. The light
rays from each image traverse different geometric path lengths and encounter different regions of the gravitational potential (Shapiro 1963). The differences in arrival
times between images (referred to as "time delays") depend on the angular-diameter
distances between the lens, source, and observer. Therefore, measurements of these
delays can be used to estimate Hubble's constant, Ho (Refsdal 1964; Refsdal 1966).
24
'This method requires that the background source have an observable parameter which
varies with time. The variable flux at optical and radio wavelengths has led to measurements of a time delay in 0957+561 (Lehir et al. 1992; Press et al. 1992). However,
the uncertainties in the lensing models result in large uncertainties in the estimate of
Ho (Falco et al. 1991; Roberts et al. 1991).
Gravitational lenses are important since they can provide estimates of astrophysical parameters. The confidence limits depend on the accuracy of the lens reconstruction. Small changes in the matter distribution lead to uncertainties in the astrophysical interpretations. MG0414 was chosen as the focus of this study because the four
image configuration gives more constraints in the lens inversion than the two image
systems, such as 0957+561. Although MG0414 does not probe the lens potential as
thoroughly as a ring system, the extended emission in the ring is too large to be
variable. Monitoring of the components in MG0414 have found them to be variable
at a level of a few percent (Moore & Hewitt 1995). In addition, this system consists
of four bright, compact components at many wavelengths, making high-resolution
observations feasible.
In this thesis, we pursued the goal of understanding the system through the lens
modeling of the 0'.1 resolution VLA observations and the 0"01 resolution Very Long
Baseline Interferometry (VLBI) observations of MG0414. Using VLA observations
of the ring, MG 1654+1346, Kochanek (1995) modeled the lens galaxy and found
evidence for dark matter in the core of the galaxy, a giant elliptical at a redshift of
0.254. The lens galaxies in gravitational lens systems are generally large ellipticals
because smaller spiral galaxies will not create the easily identified large separation
systems. Therefore, the study of gravitational lenses can provide a window into the
dynamics and structure of the cores of elliptical galaxies.
VLBI observations have proven effective in constraining the gravitational lens
systems 0957+561 and 2016+112. Porcas et al. (1981) discovered the same core-jet
structure in both 0957+561 images. Using this structure, Gorenstein et al. (1988a)
derived the magnification matrix which relates the brightness distribution in one
image to that in the other. Recent VLBI observations by Garrett et al. (1995) found
25
that the magnification matrix varies across the jet, as claimed by Conner, Lehar, &
Burke (1992). To investigate the unusual gravitational lens 2016+112, Heflin et al.
(1992) observed this system using a global VLBI array. He found the system was
not consistent with a linear magnification matrix, indicating the source is evolving
on a shorter timescale than the time delay, predicted to be between 8 and 11 months
(Narasimha et al. 1987; Narasimha & Chitre 1989).
1.3
Chapter Summary
In Chapter 2, we review the theory of interferometry and aperture synthesis, including the effects specific to VLBI. Chapter 3 reviews the basic theory of gravitational
lensing and the lens inversion problem. This chapter discusses the problems with
the current implementation of the LensClean algorithm (Kochanek & Narayan 1992)
and presents a modified implementation to remove these effects. In Chapter 4, we
extensively model the MG0414 system using the 15 GHz VLA observations (Katz &
Hewitt 1995). We explore a variety of simple lens models in LensClean and more complex models using "point solvers" (see Section 3.2). This work has been performed
in collaboration with Professor Christopher Kochanek at the Harvard-Smithsonian
Center for Astrophysics. Chapter 5 presents the 5 GHz VLBI observations of the
MG0414 system. The data reduction procedures used to achieve the final images
and their reliability are discussed. Using the components in the VLBI images, we
modeled the system with the lens model parameterization which best fit the 15 GHz
VLA observations. Chapter 6 discusses the results and future work.
26
Chapter 2
Interferometry Theory
2.1
Introduction
Any finite continuous function can be fully described by a complete set of Fourier
components (Arfken 1985; Churchill & Brown 1976). Consider a two-dimensional
brightness distribution I(x,y)
emitted from a source. Each Fourier component is
isolated by modulating I(x, y) with the corresponding basis function and integrating
over space. The Fourier component is specified by the wave-vector u + v,
which
is orthogonal to the two-dimensional wave fronts and has length equal to the spatial
frequency of that component. Therefore, the Fourier basis function is e2ri(ux+vy)and
the projected Fourier component is,
" ' (u+vy).
So(u, v) = dxdyI(x, y)e2
(2.1)
The function r(u, v), known as the complex visibility function, is the Fourier component normalized by the total flux of the source,
S, = J dxdyI(x, y).
(2.2)
Equation 2.1 relates the brightness distribution and the complex visibility function
through a two-dimensional Fourier transform, and, thus, the x-y (or sky) plane and the
27
u-v (or visibility) plane are Fourier conjugate spaces. We have assumed that I(x, y) is
defined on a two-dimensional plane rather than the celestial sphere. This assumption
is valid in the regime where the source size is small compared to the curvature of the
sphere. The brightness distributions of astrophysical sources are purely real, requiring
the visibilities to be conjugate-symmetric, i.e. F(u, v) = r*(-u, -v).
Therefore, to
fully specify a source brightness distribution, the complex visibility function must be
measured over half the u-v plane.
A measurement with an interferometer is equivalent to projecting a sinusoidal
pattern, or "fringes" (see Section 2.2), and integrating over the sky. Therefore, an interferometer measures the spatial Fourier components of astrophysical sources. Each
(u, v) point corresponds to the vector pointing from one antenna to another, where the
projected antenna separation is
u 2 + v 2 and the orientation is specified by tan - 1 ().
In the following sections, we review the basic interferometry theory as applied to
radio astronomy. For a more detailed discussion, we recommend Thompson, Moran,
& Swenson (1986) and the VLA Summer School lectures (1989).
2.2
The Two-Element Interferometer
The basic element in interferometry is the two-element interferometer. Consider the
two antennae shown in Figure 2-1. The relative orientation of the antennae is given by
the baseline vector b, pointing from one antenna to the other. The sign of the vector
is not important, since the u-v plane is conjugate-symmetric. The radiation from a
distant monochromatic point source of flux S, reaches antenna 1 before antenna 2 by
the geometrical delay,
r9
where
=
(2.3)
is the source position unit vector and c is the speed of light. To compensate
for this geometrical delay, we introduce an instrumental delay in the signal path from
antenna 1,
o=
(2.4)
-b
C
28
tg
p~~~~~to
I2
S (t)
s (t)
2
1
Figure 2-1: Diagram of the two element interferometer. b is the vector from antenna 1
to antenna 2, g is the direction to the source, r is the geometric time delay, and o7
is a delay introduced before correlation.
The effect of r,, is to "point" the interferometer at a specific position in the sky sg,
known as the delay center. Therefore, the signals received from both antennae are,
si (t) =
)
(2.5)
(t- Tg)
(2.6)
Scosw(t-
s 2 (t) = Vcosw
where w is the observing frequency. Multiplying these signals together, we find,
VR= So [Cos (2t
-
, -
) + cosw (rg -
)]
(2.
(2.7)
Since we are interested in the relative phase of the incident emission, we keep only
the latter term. As the Earth rotates, r -
m varies very slowly compared with the
2t - rT- T term. Applying an appropriate low-pass filter, we obtain the correlated
29
signal,
(2.8)
VR = So cos (27rbA. (s- s))
where bA is the baseline vector measured in wavelengths. We have scaled the response
by a factor of two since half the correlated response has been filtered out.
response is a function of position
This
and effectively projects a sinusoidal function on
the sky, known as "fringes". If the "fringe" pattern and source position are such that
bx (s'-
o) = 1/4, the correlated signal vanishes. These nulls in the response function
can be easily moved by delaying the signal from telescope one by an additional r/2w,
introducing a 7r/2 phase shift into the response function. Therefore, the response
becomes,
VI = SOsin (27rbx (s-
so)).
(2.9)
We remove the nulls in the response functions by combining both VR and VI to obtain
the total complex response function to a point source,
V = So ¢2
e i -( sx s )
g
.
(2.10)
To calculate the response to an extended source, we rely on the principle of superposition. For an interferometer, superposition states that the response to a sum of
sources is the sum of the individual responses for each source. Since we can express
the extended source as an integral of point sources, the response is an integral of
point source response functions,
V (b) = J I(Q)e2 -ibg(\ )o)dQ.
(2.11)
where I(Q) is the brightness distribution of the extended source on the celestial sphere.
We assume the individual antenna responses (primary beams) are constant over the
source. This assumption is valid when the source size is small relative to the primary
beam and the antenna elements track the source throughout the observation.
The interferometer response is not a two-dimensional Fourier transform of the
source brightness distribution because I(Q) is defined on the celestial sphere, not a
30
flat two-dimensional surface. By limiting our field-of-view to a region defined such
that d
- d2x, we approximate a small region of the sphere as a plane. This trans-
formation projects the points on the celestial sphere onto a tangent plane, where
the tangent point is so. We express the relative source coordinates s'- so using a
rectangular coordinate system defined such that the x-axis points towards local west
and the y-axis points towards local north on the tangent plane. We use (u, v) for the
baseline coordinates measured in wavelengths (i.e. b = (u, v)). In this coordinate
system, we recover the Fourier transform relationship between the response and the
two-dimensional brightness distribution,
V(u, v) =
(2.12)
I(x, y)e27ri(u+vY)dxdy.
An interferometer measures the complex visibility function, which is the ratio of the
correlated flux density V(u, v) to the total flux density,
f
F(uv)
I(x, y)e 2 ri(ux+vY)dxdy
f I(x, y)dxdy
2.3
(2.13)
Aperture Synthesis
From Equation 2.13, we see that by measuring all the Fourier components of I(x, y),
we can reconstruct the brightness distribution simply by Fourier inversion,
I(x, y) = So
J
(u, v)e-27ri(ux+VY)dudv
(2.14)
where SO is the total flux of the source. In practice, we do not have measurements
over the entire visibility plane since resources are limited.
Radio interferometers
exploit the rotation of the Earth to fill in the u-v plane. Since an interferometer
has a finite number of antennae, the u-v plane is sampled at a small number of
points (uij(t),vij(t)),
each corresponding to a single baseline at a given time. As
the Earth rotates, the orientation of a baseline changes, tracing out arcs in the u-v
plane. If the source does not vary on the timescale on the observation, the entire
31
u-v track can be used to reconstruct the image. Each measurement along the track
is an average of the true complex visibility function over a finite integration period.
We assume the integration period is short such that the measured visibility is an
accurate representation of the underlying true visibility. We address the effects of the
averaging in section 2.4.2.
Consider an East-West baseline and a source directly north ( = 900). At a given
time t, this baseline defines two u-v points: (u 12 (t), v12(t)) and (-u
12 (t),
-v 12 (t)). For
a continuous observation using an integration time r,, the rotation of the Earth causes
the baseline to trace out a full circle over a 12 hour period, where half the circle is
made up from (u 12 (t), v 12(t)) and the other half is made up from (-u
12 (t),
-v 12 (t)).
This u-v track consists of samples separated by a distance,
rul
2 (t)
1+v
2 (t)
(i2 hirs)
(2.15)
In general, the projected baselines will traces out ellipses in the u-v plane.
An interferometer consisting of N stations measures r(u, v) on N(N-1)
baselines
simultaneously. We can express the measured visibility m(u, v) as the product of
the true visibility r(u, v) with a sampling function,
rm(u, v) = F(u, v)S(u, v)
(2.16)
where the sampling function is,
Nt
N
S(u, v) = E s wijs(u
k
-
uij(tk))S(V - Vii(tk))
(2.17)
i,j
and Nt is the number of integration periods and wij are weighting factors to adjust
the shape of the instrumental response function. The effect in the image plane is to
convolve the true image by the instrumental response,
Id(X, ) = I(x, y) 0 B(x, y)
32
(2.18)
where Id(x, y) is the dirty image and the dirty beam B(x, y) is the Fourier transform
of the sampling function. Typically, the dirty beam contains many lower level peaks
(referred to as "sidelobes") separated from the main central peak. For sparse arrays,
the heights of the sidelobes can be a significant fraction of the height of the main
central peak.
The resolution of the interferometer is characterized by the full-width at halfmaximum (FWHM) of the dirty beam central lobe. The exact size and shape of the
dirty beam depends on the distribution of visibility samples in the u-v plane and their
weights, given by wij. In the absence of u-v tapering, the FWHM is approximately
determined by the angular distance which causes a 27r phase rotation on the largest
projected baseline b,
OFWHM
ba\
-
b
(2.19)
where A is the observing wavelength. In A-configuration, the VLA has a maximum
baseline length of _ 40 km. Therefore, a A2 cm observation would have a minimum
beam size of ~ 0'1.
We see that largest baseline length in an interferometer is
analogous to the diameter of a filled aperture telescope in determining the resolving
power of the instrument.
Since we can reconstruct images of astronomical sources
without measurements throughout the entire aperture, this technique is also known
as aperture synthesis.
The weighting factors wij are used to modify the shape of the dirty beam. The
two commonly used weighting schemes are natural weighting and uniform weighting
(Sramek & Schwab 1989). Natural weighting applies the same weight to all the data
points, yielding the highest signal-to-noise ratio. Since the shorter spacings spend
more time in a given region per unit area, natural weighting tends to emphasis the
short baselines and broaden the dirty beam, resulting in lower resolution. Uniform
weighting assigns weights which are inversely proportional to the number of data
points within a certain region,
Wi= N
(2.20)
where Nij is the number of data points within a region centered on uij. Typically, the
33
region is defined as a square box (referred to as a u-v box). The shorter baselines are
more densely populated and thus down-weighted with respect to the longer baselines.
The result is that uniform weighting has a smaller beam than the naturally weighted
beam at the expense of a larger average sidelobe level.
2.4
Smearing Effects
2.4.1
Bandwidth Smearing
The discussion up to now has assumed a monochromatic signal. In practice, the
source is broadband and measured across a finite bandwidth.
The non-zero band-
width has the effect of decorrelating the measured signals across the bandpass (Bridle & Schwab 1989). The response to a frequency-dependent brightness distribution,
I(x, y, v), is,
V(u,v) = dxdy
[
dvI(x, y, v)e2ri(S-S)v/c]
J
(2.21)
where the bandpass is assumed to be rectangular, centered on v0 , and has width Av.
If we assume the cosmic signals vary slowly with frequency across the bandwidth
of the receiver, then the brightness distribution can be approximated as a constant
function evaluated at the center frequency. Integrating over frequency, we find,
V(u, v) =
where r = b (-
so)/c.
dxdyI(x, y, Vo)e2i a
o( - so)
[sin(7rAv)]
(2.22)
We see that the brightness distribution is multiplied by a
sinc function, limiting the field-of-view. The amount of sky which can be mapped is
given by when the sinc function reaches its first null,
C
O
=
b1 Av
34
(2.23)
where Ozx is the width of "delay" beam and by is the length of the projected baseline.
The VLA has a bandwidth of 50 MHz resulting in a delay beam of _ 30" for a
maximum baseline length " 40 km (A-array). If we examine the effect in the visibility
plane, we see that the bandwidth corresponds to the radial width of the u-v tracks,
which is integrated to yield the measured visibility point. If the visibility function
varies rapidly over that width, then the measured visibility point does not accurately
represent the true visibility function. In the measured image, the effect smears the
source along the radial direction. The amount of smearing depends on the distance
from the delay center in the sense that larger u-v distances correspond to larger
distortions. Bandwidth smearing is serious because the effect modifies the convolving
function to one which is a function of position on the sky. Deconvolution algorithms,
such as CLEAN and NNLS (see section 2.5), assume the dirty beam is independent
of position and will create spurious extensions due to this type of smearing.
2.4.2
Time-Average Smearing
Time-averaging of the visibilities causes an effect analogous to bandwidth smearing
except in approximately the orthogonal direction (Bridle & Schwab 1989). Each
measured point in the u-v plane is an average of the visibilities over some integration
period. The (u, v) point assigned to the integration period is the midpoint of the
integration interval along the u-v track. In exactly the same manner as bandwidth
smearing, if the visibility function varies rapidly over the integration period, the integrated visibility will not be an accurate representation of the true underlying visibility
value. Clearly, the longer baselines are more affected since, for a given integration
time, they trace out a larger distance in the u-v plane (see Equation 2.15). Therefore,
the time-average smearing effect is baseline dependent, resulting in a smearing of the
dirty beam as a function of position.
Although this effect does not alter the flux detected in the image, the amplitudes
in the image are reduced due to the smearing of the dirty beam. For a source which is
a distance 0 away from the phase center, the average fractional reduction in amplitude
35
for an integration interval of Ta is,
(Rr.) =
-
w
12
( OFWHM )
where w, is the rotational frequency of the Earth,
(2.24)
FWHM is the half-power beam-
width of the synthesized beam, 0 is the offset of the source from the phase center,
and a is a constant of order unity describing the details of the baseline distribution
and tapering function used (Bridle & Schwab 1989). The amplitude of the smearing
depends on the ratio of the source separation to the beam width and, therefore, affects
VLBI observations more seriously. If we require that (R-) 2~ 0.99 for an averaging
period of Ta = 30 s, the extent of the source must be A 2 0 0 0FWHM in diameter.
2.5
Image Reconstruction
Radio interferometers use sparse arrays of antennae to achieve high-resolution observations of cosmic sources. The resulting dirty beam will contain sidelobes away
from the main peak. The amplitude of those lobes is determined by the amount of
the u-v plane which is filled during the observation. Equation 2.18 shows that the
result of choosing these sparse configurations is that the measured image is strongly
corrupted by the convolution with the dirty beam. Even for densely sampled VLA
observations, significant sidelobes are present away from the central peak, introducing
correlations between widely separated regions in the dirty image. In addition to these
sampling effects, the measured visibilities are contaminated by errors in the observation. The primary sources of the errors are antennas-based, such as atmospheric
and instrumental effects. Baseline-based errors are present due to errors such as correlator problems; however, we ignore these since these effects are usually far smaller
than the antenna-based errors. For interferometers with N > 3 antennae, there are
N(N- 1)/2 > N measured complex visibilities. Therefore, by constraining the errors
to be antenna-based, we need only solve for N amplitude and N phase errors, given
N(N - 1)/2 complex constraint equations.
36
2.5.1
Self-Calibration
The noise is characterized by an amplitude and phase error for each antenna. Since
these errors are due to effects such as clouds, they are allowed to vary with time. Let
ci(t) be the multiplicative complex error associated with antenna i at time t. Then,
the measured visibility on the baseline from antenna i to antenna j is,
Vm(Uij) = Ci(t)ej(t)V(uij)
(2.25)
Let Im°del(p) be a model of the true image. By Fourier transforming this model,
we can directly compare the measured visibility with the predicted model visibilities
Vmodel(U).
If a reference antenna is chosen to have zero error, the errors for all the
other antenna can be solved for using the constraints,
E (tj(t
=
V (ufj(t) )
(2.26)
Since we have arbitrarily defined the zero point, the absolute position (phase) information is lost. The flux reference is preserved by requiring the mean amplitude
correction to the visibilities be unity.
Since an a priori model of the source is rarely available, an iterative approach to
image reconstruction is used to account for the effects of noise and discrete sampling.
Self-calibration (Cornwell & Fomalont 1989) solves for a model of the true image
and the antenna-based errors by applying consistency requirements on the data. At
each iteration, the visibility data is Fourier transformed to create the dirty image.
Deconvolution methods, such as CLEAN and NNLS, are used to reconstruct a model
of the true image. Using Equation 2.26, the antenna-based errors are estimated and
the visibilities are corrected by,
Vcorr(uij) = gS(t)gj(t)V m (uij)
where g(t) =
(2.27)
/li(t) are the antenna gains. These iterations are repeated until a
self-consistent solution for the gains and image is found. Typically, amplitude errors
37
in VLA observations are A 2 percent and not usually corrected. On the other hand,
VLBI observations can have amplitude errors of 10 percent or more. The variations in
the phase error are usually dominated by changes in the atmosphere and can exceed
27r.
This method of image reconstruction raises the question of uniqueness. In most
cases, poor choices of the self-calibration or deconvolution parameters leads only to
solutions in which the solution is not optimal. Only on rare occasions-extremely
poor u-v sampling with very few antennae-has
self-calibration produced multiple
significantly different solutions (Cornwell & Fomalont 1989).
2.5.2
Deconvolution Algorithms
The choice of algorithm is important since the deconvolution techniques are not perfect and can introduce systematic errors into the reconstructions.
Three deconvo-
lution algorithms are in use in radio astronomy: CLEAN, NNLS, and MEM. The
most frequently used algorithm is the CLEAN algorithm (H6gbom 1974; Clark 1980;
Schwab 1984), which is based on the assumption that sources can be decomposed
into point sources. In order to implement CLEAN efficiently, each visibility point
is convolved and sampled onto a 2 x 2 grid in the u-v plane (typically referred to
as "gridding"), so the Fast Fourier Transform (FFT) routine can be used. The u-v
grid dimensions-both
the cellsize and number of pixels-should
be such that, in the
image plane, the dirty beam is oversampled (i.e. the width of the dirty beam is 2 3
image plane pixels).
The normal implementation divides the procedure into the minor and major cycles.
During the minor cycle, the algorithm finds the position and flux of the peak in the
dirty image. At this position, the dirty beam is subtracted from the dirty image, scaled
to a fraction (called the loop-gain) of the peak flux and a point source with the same
flux is added to the clean map. The procedure is repeated until a stopping criterion
is satisfied, typically when all the pixels are less than a fixed fraction of the dirty
image peak at the start of the minor cycle. In the major cycle, the accumulated clean
components collected in the previous minor cycle are subtracted from the ungridded
38
complex visibilities. When a specified number of clean components are subtracted
or the maximum residual in the image is below a fixed value, the algorithm stops.
The clean component model is used as the model for the true image structure in selfcalibration. Since spatial frequencies much larger than those sampled by the longest
baseline are not well constrained, we remove these frequencies through convolution
before producing the final image. The convolving function (referred to as the clean or
restoring beam) is a gaussian which is fit to the central lobe of the dirty beam. The
clean image is created by convolving the final reconstructed image-the
clean components-with
collection of
the clean beam. The residual image is added to include any
flux which was not reconstructed in the deconvolution process.
A number of problems are intrinsic to this method of deconvolution.
CLEAN
is an inherently "local" algorithm in that each new clean component cannot alter
any previously subtracted component. This independent nature leads to errors in
both compact and extended structures. CLEAN subtracts from the peak of the dirty
image, but due to noise and beam sidelobes, the peak is not the actual location of
the underlying compact source. CLEAN compensates for the initial error by adding
many nearby low level components.
This behavior adds spurious "skirts" around
bright components, and it is characteristic of CLEAN reconstructions of compact
components. Another well-known problem affects extended sources, where CLEAN
tends to subtract in periodic "stripes" with a period equal to the distance between
the beam peak and the first negative sidelobe (see Cornwell 1983, for example).
Briggs (1994) has adapted the Non-Negative Least-Squares (NNLS) algorithm
(Lawson & Hanson 1974) to the deconvolution problem. NNLS generalizes CLEAN
to use a least-squares fit to the visibility data with a finite number of non-negative
clean components. The algorithm constructs a model for the source by minimizing the
mean-squared difference between the dirty image and the convolution of the source
model with the dirty beam,
= J d2 (B ® Im°del( ) - Id(7))2
39
(2.28)
where Im°del(7) is the model of the true source distribution. At each iteration, the
model adds another component to the image, located at the pixel which produces
the largest reduction in Ao.The fluxes for all the components in the model, including
the newly chosen component, are simultaneously optimized by minimizing
in a
least-squares sense with the requirement that all fluxes remain non-negative. The
algorithm converges when adding components does not reduce pOor when all the
pixels in the image have been included. As with CLEAN, the clean component model
is used as the self-calibration model of the true image. To produce the final NNLS
image, the components are convolved with the restoring beam and added to the
residual image. The NNLS algorithm is "global" in the sense that the correlations
between clean components are accounted for in the flux optimization step. NNLS
also enforces positivity in the reconstructed source by constraining each pixel to be
non-negative. The NNLS algorithm performs far better than CLEAN on partially
resolved sources (Briggs et al. 1994), but NNLS is both a memory and time intensive
calculation. It requires storage and inversion of a dense matrix with NdNs entries,
where Nd is the number of pixels in the dirty image and Ns are the number of source
pixels in the reconstructed image. Since matrix inversion scales in time approximately
as O(NdN2), this algorithm is only beginning to become computationally practical.
Unlike CLEAN and NNLS, the Maximum Entropy Method (MEM) (Narayan &
Nityananda 1986, for example) does not represent the source as a collection of point
sources. MEM attempts to reconstruct the image under the constraints that the
model is positive, fits the visibility data, and maximizes the image "entropy". MEM
performs well for sources with diffuse emission or simple structure, but poorly reconstructs compact sources embedded in extended emission (Cornwell & Fomalont 1989).
The nature of the entropy function leads to reconstructions with edges only as sharp
as the data requires. This "smoothness" constraint can cause spurious extensions in
the reconstructed images of compact sources.
40
2.6
Very Long Baseline Interferometry
Historically, Very Long Baseline Interferometry has used isolated single-dish telescopes around the world as elements of a "global" interferometer. Differences in the
atmospheric conditions and instrumentation at each site cause large differences in the
amplitude and phase of the electric field recorded at each site. Furthermore, in contrast to the VLA, the independent time standards at each site are not phase-locked,
so errors in the "clocks" corrupt the observed phases. These effects must be corrected
before the data can be imaged.
2.6.1
Amplitude Calibration
To obtain correct amplitudes, the normalized visibilities are scaled by,
ysTTsys
V(uij) = b
GiGj
(2.29)
where TS8Sand Gi are the system temperature and gain of antenna i. The digitization
losses are characterized by the parameter b. For 1-bit sampling, b is 1.57 (Thompson
et al. 1986). The gains are computed using one of two sources: established gain
curves or antenna temperature measurements. The first method relies on observations
of calibration sources. In this context, a sufficient definition for the calibrator is a
compact source of known flux. Many observations of these calibrators can be used
to map out the gain of the station as a function of elevation. Most telescopes use
altitude-azimuth mounts, so the gain is parameterized by only the elevation angle.
Some stations, such as the NRAO 140 foot telescope in Greenbank, use polar mounts
which require the gain to be a function of both hour angle and declination,
The second method relies on antenna temperature measurements taken during the
course of the observation. The antenna temperature is a measure of the total power
received due to the source. The measurement is performed by recording the response
41
of the antenna both on and off source,
T t = T-
T~F
(2.30)
This procedure, known as "on-offs", requires knowledge of the flux of the source of
interest. The gain of the station i is then given by,
Gi =
Tant
2.6.2
S
(2.31)
Fringe Fitting
This discussion draws heavily from Schwab & Cotton (1983). When the antenna
signals are correlated, a model of the visibility phase is subtracted. This correlator
model accounts for the array geometry, atmospheric delays, and constant clock offsets.
However, differences in the independent time standards along with uncertainties in
the antenna positions and atmospheric propagation delays cause significant errors in
the observed phase. Therefore, the VLBI signals are correlated with several different
time lags between each station. The observations are also simultaneously recorded
at several different frequencies. Let the observed visibility be
rIT(fk,
rt), where fk
is the kth frequency channel and rl is the Ith correlator delay for baseline ij. We
can Fourier transform rF(fk, r ) into the time and frequency domain to obtain the
observed cross-spectral function, rF'(tk, vl), which is related to the true cross-spectral
function Vij(tk, vl) by,
r3 (tk, vL) = gi(tk, V)gj*(tk, lI)Vij(tk, vl) + Eijkl
(2.32)
where the antenna-based errors are absorbed into the complex functions,
gi(tk, VI) = aieii(tkVI)
(2.33)
and the amplitude ai(tk, vl) is assumed to vary slowly compared to the antenna phase
Oi(tk,
vi). The baseline-based error
ijkl
is assumed to be small and neglected. We
42
can expand the observed visibility about (to, vo) to first order,
ri'(tk,V)
aiajVij(to, Vo)exp {i [(oi - 0j)(to, vo)]}
(4i-4 j +
wij)
(tk- to)
(to,vo)
+i
Il,,
a"
)(·l -I0)
(2.34)
where ij is the phase due only to the source brightness distribution. We define the
fringe delay (or "delay") as,
rij o= a(¢i
- -V++
+
(2.35)
(to,vo)
(to vo)
and the fringe rate (or "rate") as,
rij-
+ ij)
at l(to,vo)
-j
(2.36)
tj
The delay is the difference in the clock errors and the rate is the difference in the
derivative of the clock errors.
The traditional baseline-based method of fringe fitting solved for the delays and
rates by Fourier transforming the distribution,
6(t- (tk - to))(v - ( - Vo))PiJ(tk,v)
Fij(t, v) =
(2.37)
kl
to obtain Fij(r, r) and searching for the maximum. The observed visibilities are then
corrected by,
rcFrr(tk, V1) = rF (tk, vl) exp {-i [(tk - t)ri
+ (vk - vl)rij]}
(2.38)
where rij and rij are the delay and rate solutions corresponding to the peak. Note that
the antenna phase
/i'
is not removed, so the visibilities are corrected up to a constant
phase offset for each antenna. However, since these offsets are antenna-based, they
43
are removed during self-calibration. The major drawback of this method is that the
source must be strong enough on each visibility so that the delays and rates can be
solved.
The global fringe fitting procedure has been developed to solve for all the delays and rates in an antenna-based framework similar to self-calibration (Schwab &
Cotton 1983). We can define the antenna-based phase, delay, and rate by,
Oi(to,vo)
i =
(2.39)
v (to,vo)
=
iat (tovo)
where the delay and rate are no longer functions of the true phase of the source bij.
Let Viidel be a model of the true source visibility Vij. Global fringe fitting finds the
set of antenna-based parameters such that,
Fi(tk,
VI) m aiajViOdel (to, Vo)Eij,kl
(2.40)
where all the antenna-based effects have been grouped into,
Eij,kl =
exp {i [(bi-
4j)+ (ri -
rj)(tk - to ) + (ri - j)(VI - o)]}.
(2.41)
Since the parameters enter only through pairwise differences, we define a reference
antenna which has zero error (i.e. Oref
rref
=
= Tref =
0).
Since we are interested only in the visibility phase, we assume the model amplitudes are equal to the calibrated amplitudes. In other words, we assume that after
dividing the data Vmby the model
the result is a complex phase
Vmodel,
where q is the observed visibility phase and
°
q model
is the phase of
ei(
-
model),
Vmodel.
Consider
12de(tk,
V
1 ))]}
the distribution for a baseline from antenna 1 to antenna 2,
F12 (t, v) = E S(t - (tk -to))6(
kl
- (v -
vo))exp{i [(1 2 (tk, vl)-
(2.42)
44
If we choose antenna 1 as the reference antenna (l
1
= r = rl = 0), then by Fourier
transforming F 12 (t, ) and searching for the peak, we can solve for r 2 and
addition, the phase of F1 2 (t,
VI)
gives
iP 2 .
2.
In
For antennas 1, 2, and 3, we can define the
closure relation,
¢12- (~ de ' = (13 -
d el)
-mo
-
- Ome )
°
(2.43)
and, therefore, solve for the parameters for antenna 3. Then, the closure relations
for antennas 2, 3, and 4 can be used to find the antenna 4 parameters, and so forth
until all the antenna parameters have been determined. This algorithm reduces the
requirement that all the baselines have signal-to-noise ratios large enough to solve for
the delays and rates to the requirement that at least one baseline to each antenna have
a large signal-to-noise ratio. Moreover, since the problem is solved assuming antennabased effects, the closure relations cannot be corrupted through this procedure.
The global fringe fitting algorithm implemented within the AIPS data reduction
system incorporates the information from multiple baselines to a single antenna rather
than the single baseline function F12 (t, v), giving better parameter estimates (Cotton
& Schwab 1993; Rogers 1993a; Rogers 1993b).
45
Chapter 3
Gravitational Lensing and Lens
Inversion
3.1
Gravitational Lens Theory
Gravitational lensing occurs when the light rays from a distant source are deflected
by an intervening massive object. The effects of the gravitational potential are adequately described in the weak field limit of general relativity (Refsdal 1964). Since the
distances between the lens, object, and observer are a significant fraction of the universe and the interaction region is approximately given by the size of a galaxy, we can
describe the system using the thin lens approximation. The photons are unaffected
until they reach the lens, at which point, their trajectories are abruptly changed by
a two-dimensional surface mass density. This scenario is illustrated in Figure 3-1.
Assuming small angles, we derive the lens equation,
-=--°D
at
(3.1)
where Dij is the angular diameter distance from point i to point j (i < j) (Weinberg 1972). The positions vectors 0 and / are two-dimensional vectors defined in the
lens (image) plane and the source plane, respectively. The lensing equation describes
the mapping from the image plane 0 to the source plane /.
46
For a point source of
Apparent Image
True
)bserver
Ak-A~
Plane
>
> Lens
Source
Plane
DLS
DOL
Dos
Figure 3-1: Gravitational lens diagram. The path of a single light ray is drawn. Dij
is the angular diameter distances from point i to point j, 0 is the angular position of
the image (apparent source), d is the angular position of the source in the absence of
a lens, and (Et is the deflection angle.
mass M, the deflection angle is given by (Schneider et al. 1992),
4GM
at = c2D
0
(3.2)
c2DoL112
Using superposition, we extend this definition to find the deflection due to distributed
mass,
at
4G
C2DoL
d2(t
)
(0 - 0')
-
(3.3)
where E(0') is the surface mass density of the lens. The two dimensional gravitational
potential due to E(9') is,
t
()
=
4Gc d2
() In
1-
I.
(3.4)
Beginning with the gravitational potential, we can invert Equations 3.3 and 3.4
47
to recover the deflection angle,
at = VDoL
t
D
ODOL Oe
(3.5)
and the mass distribution,
V7DoLObQt =
(3.6)
4rG(0).
To remove the dependence on the distances, we define dimensionless counterparts for
the potential,
b=
S
t
(3.7)
St = V
(3.8)
DOLDos
and deflection angle,
Dos
so the lens equation becomes,
= - .
(3.9)
The Laplacian of the dimensionless potential yields twice the surface mass density in
units of the critical density,
V
(3.10)
= 2 s
where CE = c2 Dos/47rGDoLDLs.
For cosmological lenses, the distances between the observer, lens, and source extend over large fractions of the observable universe. Therefore, the curvature of the
universe must be taken into account. In a Friedmann cosmology with A = 0, the
angular diameter distance between a point at redshift zi and a point at redshift zj
(zi < j) is,
2c (1 - Qo)(Gi- Gj) + (GiG2 - GGj )
D
where Gi = Z, 1 +
Ho
2(1 + z)(1 + z)
2
Ho is Hubble's constant, and Qfois the ratio of the density
of the universe to the critical density (Blandford & Narayan 1992). The Friedmann
cosmology assume the universe is homogeneous and isotropic. On the distance scales
which we are interested, approximately a gigaparsec, the universe appears uniform
48
and is approximately described by a Friedmann cosmology (Seljak 1995). The DyerRoeder equation can be used to determine the angular diameter distance as a function
of redshift in a universe with gravitationally bound objects (Dyer & Roeder 1973;
Schneider et al. 1992). This equations assumes that some fraction of the mass in the
universe has coalesced into gravitationally bound objects, all of which are far from
the bundle of light rays of interest.
4
I
I
I
I
1
a
.2
.
-4
.2
0
.
I
2
I
4
0[0E]
Figure 3-2: The deflection angle diagram for a point lens of mass M. The axes are
in units of the "Einstein" ring radius.
The characteristics of a lensing system can be seen by plotting a and 0 -
as
functions of impact parameter 0. The image positions are found at the points where
the two curves intersect. Figure 3-2 shows the lensing behavior for a point mass lens.
Since
is a two-dimensional vector and the mass distribution has circular symmetry,
we only plot the
at a given position angle. The diagonal lines correspond to 0- /3
for different values of /. It is clear that for any /3, there are two and only two images.
For large ,, one image lies near the source position and one image lies near the lens
center. For smaller values of /, the two images tend towards the same distance from
the lens center. Since this lens model is axially symmetric, a point source directly
49
behind the lens will be imaged into an "Einstein" ring with a radius,
OE =
/4GM
VCI
DLS
DosDOL'
(3.12)
0o
-2
-4
-2
-4
0
2
6
Figure 3-3: The deflection angle diagram for a general lens with circular symmetry.
For a general lens with circular symmetry, the deflection angle does not diverge at
small values of /. Figure 3-3 shows a deflection angle diagram for an extended lens
and four source positions:
A,
PB, 3c, and PD. As 0 becomes large compared to the
size of the lens, the mass distribution is more accurately approximately by a point
mass. Therefore, far from the lens, 5 falls off as 0- 1. The behavior of a for small
impact parameters depends on the exact mass distribution in the lens. The source
positions PA and fOcdefine the boundary of the multiply-imaged region for this lens.
Within this region, each source maps into three images, corresponding to the three
points of intersection. Curve 0'-
B describes a source directly behind the lens center
which is lensed into three images, one of which is located at the lens center. We see
from the steep rise in the deflection function, any multiply-imaged source will have
one image near the lens center. As a source crosses from the triply-imaged region to
50
the singly-image region, two of its images merge together and disappear. Although
the more distant sources, such as /D, are shifted and distorted, they do not map to
multiple positions in the image plane. We see that a circularly symmetric lens can
produce at most three images. Hence, the observed lens systems, such as MG0414,
require the lens with at least a monopole and quadrupole moment.
The magnification of each image can be found by considering the effect of the lens
on a bundle of light rays emanating from the source. In the absence of the lens, the
received flux from a source of surface brightness Iv is Idf,
where dQ is the solid angle
subtended by the source. Since gravitational lensing conserves surface brightness, the
magnification is due solely to the distortion of the solid angle,
d2ens
dlens
where dIens
(3.13)
is the solid angle subtended by the distorted image. We describe the
undistorted area as a function of the distorted area using the mapping given by the
lens equation (Equation 3.9),
dQ =
00
dQilens
(3.14)
indicates the determinant. Therefore, the magnification is,
where 11
P
=
00
(3.15)
001
1z
where I is the identity matrix. In a frame where the magnification matrix,
M=
00]
51
[
(3.16)
is diagonal, the magnification is simply,
=
where Ai is the ith eigenvalue of M.
A1
0
0
A2
0=
A1 A2
(3.17)
For typical quasar-galaxy lens systems, the
average magnification is - 10. If the magnification is negative, then the distortions
include a reflection through the axis corresponding to the negative eigenvalue. In this
case, the image is said to have "negative" parity. A positive magnification means the
image has "positive" parity.
The magnification of a point source is,
=132
+
/32
4
2
(3.18)
where the + solution corresponds to the image on the same side as the source and
the - solution corresponds to the image on the other side of the lens (Schneider
et al. 1992). As the source positions grows large, the magnification of the + image
approaches unity and the - image dims to zero. As / - 0, the magnification diverges
and at p = 0 the source is distorted into an entire ring.
For a general elliptical lens, the magnification is infinite at two loci of points in
the source plane. Each curve corresponds to locations where one of the eigenvalues
of the magnification matrix diverges. These curves (or "caustics") define regions in
the source plane which are mapped into regions in the image plane with different
numbers of images. Figure 3-4a show the caustics for a generic non-singular elliptical
lens. The outer (or "radial") caustic separates the 1 and 3 image region, and the inner
diamond-shaped (or "tangential") caustic separates the 3 and 5 image region. We
can map the caustics to "critical lines" in the image plane, shown in Figure 3-4b. The
radial caustic maps to the inner critical line and the tangential caustic maps to the
outer critical line. The 0, , and x symbols represent different lensing configurations
with one, three, and five images. In the source plane, the symbols represent the
sources relative to the caustics, and, in the image plane, the symbols represent the
52
0
0
X
S:X
x
Image Plane - Critical Lines
Source Plane - Caustics
Figure 3-4: The caustics and critical lines for a strong non-singular elliptical lens.
Panel (a) shows the source plane caustics. Panel (b) shows the image plane critical
lines. The , , and x symbols represent sources in the one, three, and five image
regions, respectively.
positions of all their associated images. The three-image configuration () is seen in
lens systems like 0957+561, and the five-image configuration (x) are seen systems
such as MG 0414+0534 and PG 1115+080 (Weymann et al. 1980). We see that the
"odd" image is always located near the lens center, given by the origin in Figure 34. The magnification of this image depends strongly on the size of the core radius
in the sense that a smaller core radius corresponds to a fainter "odd" image (see
Section A.2). The lack of "odd" images in many observed lensing systems indicate
that galaxies tend to have small or zero core radii.
An important observation quantity is the time delay between the images in a
multiply-imaged system. For a single light ray, the lens delays the arrival time at the
observer by,
T(9,/ _ (1Z)+
( D os) [2 (±-0-+c+()
53
onstant
(3.19)
with respect to the arrival time in the absence of the lens (Schneider et al. 1992),
where ZL is the redshift of the lens and c is the speed of light. The two contributions to the elapsed time are due to the geometric path length and gravitational
potential of the lensing galaxy. The deflection of the photons extends the length of
the trajectories needed to reach the observer. The photons also traverse through the
gravitational potential of the lens which leads to a contribution through time dilation.
The difference in arrival times between two images of a common source is,
)DOLDOS
1 [(0l 2 ( 2-
Atl2= (1+ ) (
)]-[
(&)-_ ( 2)]}
(3.20)
This relationship includes an overall distance scale for the system, D = DoLDos/DLs.
Using Equation 3.11, we find,
D - 2c (1- fQ-GL)(1 - GL)(1Ho
where Gi = /1 Fz
+
i.
Q2(1 + ZL)(1 -
- Gs)(1 - Gs)
- GLGS)(GL - Gs)
(3.21)
If the system is described reasonably well by a lens model,
Hubble's constant can be found by measuring the time delay between images.
3.2
Lens Inversion
The lens inversion problem is one of reconstructing the lens and unlensed source
from the distorted image. In general, reconstructing the underlying source altered by
some corrupting medium is not possible. In gravitational lens systems, the multiple
imaging effect adds stringent constraints allowing both the lens mass distribution and
source brightness distribution to be recovered.
A simple "point solver" algorithm for lenses with compact sources assumes all the
images map back to a common point in the source plane. The lens is parameterized
and used to map each image back to the corresponding source point. The lens parameters are determined by minimizing the discrepancy in the source positions and
fluxes. This procedure is straightforward in that the modeling constraints are obvious, i.e. requiring the images map back to a common source point. However, most
54
lens systems have some extended emission, which we would like to incorporate into
the lens model. The problem lies in defining the constraints on the modeling, since
we usually do not know in advance which regions in the image are correlated through
the lens. In addition, the finite resolution of real observations compromises the lens
reconstruction, since the image pixels do not represent the underlying image.
Kochanek & Narayan (1992) developed the LensClean algorithm as a general
tool for inverting gravitational lens systems in the presence of finite resolution and
noise. LensClean determines both the lens model parameters and source structure
simultaneously using an optimization procedure. LensClean deconvolves the image
assuming some lens model to determine the residual error in the fluxes due to inconsistencies between the data and the lens model. The error is minimized by adjusting
the lens model parameters.
The LensClean algorithm uses the CLEAN algorithm
(Hgbom 1974; Clark 1980; Schwab 1984), modified to include the distorting effects
of the lens galaxy. Since the lens completely determines the image of a given source,
LensClean iteratively reconstructs the unlensed source and compares the distorted
image of that source with the observed data.
In a manner identical to CLEAN,
LensClean constructs the source from a collection of point sources (clean components).
The image of a source plane clean component S will consist of N distorted copies
of S at positions
k with magnifications Mk. After subtracting the clean component
from the image, the mean square residual is,
2
=/
d2
Id(X)
-
f
Z IIkIB(
-
k(3.22)
k=1
where f is the source plane flux of S. By requiring that c2 be minimized with respect
to f and k, the optimum position for S is given when,
ENEjz[o
Ii II jl(B 0XB)(F,
B)(i 2Hi(B
- _j)
55
(3.23)
is maximized and the optimum flux at that position is,
f =
(3.24)
k=l
Z=l
E N=
1 J/ilj(B 0 B)(X,i - xj)
In the absence of a lens, the optimum component is the peak of the dirty image, and
LensClean reduces to the normal CLEAN algorithm.
For a fixed lens model, LensClean iteratively subtracts clean components from the
dirty image. Unlike CLEAN, LensClean will not be able to subtract all the flux from
the image because of inconsistencies between the measured data and reconstructed
image assuming a lens model. During the cleaning stage, LensClean monitors the
peak residuals after each major cycle. If the peak does not decrease by more than
one percent from the last major cycle, LensClean assumes the cleaning cycle has
converged. Errors in the lens model will create errors in the final residual image.
Therefore, LensClean uses the dispersion in the residual image as a measure of the
goodness of fit.
Each iteration of LensClean requires a knowledge of all positions and magnifications of the image plane clean components which map to the same source component, i.e. a solution to the nonlinear lens equation. Kovner (1987) showed that the
quadrupole lens models produce the same image configurations and are much simpler
to handle than truly elliptical lens models. Therefore, the lens models implemented in
LensClean include only the monopole and quadrupole moments. At least five parameters are required to specify this type of lens. Two position parameters are needed
to specify the lens center. One parameter is needed to specify the overall mass (or
critical radius) of the lens. The quadrupole terms requires at least one parameter for
its strength and one parameter for its orientation.
LensClean optimizes the lens model by minimizing the residuals as a function of
the lens model parameters, using the "downhill simplex method" (Press et al. 1988;
Nelder & Mead 1965). The "error surface" is the minimized rms residual as a function
of the lens position. In other words, the "error surface" is found by minimizing all
the parameters except the lens position. Typically, the "error surface" is corrupted
56
by systematic errors and noise, creating a lumpy texture. We outline the following
optimization procedure to prevent converging within a local minimum.
If we fix the value of the lens position, the critical radius and quadrupole parameters are well constrained because of the four image geometry in MG0414. Therefore,
we grid search over plausible lens positions (near the center of the lensing system). At
each point (lens position) in the grid, we use a "point solver" algorithm to determine
starting values for the critical radius and quadrupole parameters. Then, we optimize
only these parameters keeping the lens position (and possibly other parameters such
as core radius) fixed. We manually examine the "error surface" to find the location
of the global minimum. Then, we perform a full minimization allowing all the parameters including lens position to vary with the starting point of the optimization
near the global minimum.
3.3
Clean map LensClean
Although the LensClean algorithm is designed to account for both deconvolution and
gravitational lens effects, this method has only been applied to clean maps and beams.
We refer to this implementation as the Clean map LensClean (CLC) algorithm. This
approach is computationally simpler because it manipulates the final images instead
of the visibility data, thereby eliminating problems such as gridding and assigning
weights to the visibilities. The gaussian restoring beam is significantly smaller than
the dirty beam, reducing the computation time for the minor cycle subtractions. CLC
does, however, assume that the systematic errors in the reconstructed clean maps do
not impact the lens modeling results.
The goodness-of-fit of the lens model is measured by comparing the rms residual
in the multiply-imaged region to that of noise multiplied by the size of the multiplyimaged region in beam areas (Kochanek 1995),
Xclc
2
c~c
Nu
Ax22 (mul
27ro,2
57
or,
J
(2
(3.25)
where Nmu,, and amul are the number of pixels of linear size Ax and rms residual in the multiply-imaged region, respectively. The area of a beam is 27rab and
=FWHM'/v8/1n2.
Ob
The noise in the data ao is estimated by the rms value in
empty regions of the image. The error estimate should be limited to the multiplyimaged region, since the singly-imaged region contains no extra degrees of freedom
(Wallington & Kochanek 1995; Kochanek 1995). Since the X2 surface near the minimum may be dominated by systematic errors, a conservative scheme is used to determine the confidence levels on the parameters. First, to offset any overall systematic
bias, the X2 surface is rescaled such that the minimum corresponds to the number of
degrees of freedom, approximately given by the number of beams which fit within the
tangential critical line minus the number of parameters required by the lens model
(Kochanek 1995),
Ndof
Atan
M
(3.26)
Second, the confidence level on a parameter is defined as the largest variation of
that parameter from the best fit value with Ndof (X2
-
X)
/xin
15.1, normally
corresponding to the 99.99 percent confidence level. However, Kochanek (1995) suggests that the parameter limits should be conservatively interpreted as 95 percent
confidence levels to compensate for the unknown systematic errors in the inversion
procedure.
The final reconstructed image from LensClean is obtained by convolving the clean
components with a gaussian restoring beam. If the lens model fits the data then
the residual image contains only noise. However, if the model does not fit perfectly,
the residuals are the noise plus the difference between the observed data and the
reconstruction. Therefore, we do not add the residuals to the final image.
1
FWHM is the Full-Width at Half-Maximum of the restoring (CLEAN) beam.
58
3.4
Negative Clean Components
The LensClean algorithm has originally been implemented with the freedom to subtract both positive and negative clean components, in the same fashion as CLEAN.
LensClean treats each clean component as representing a small region of the source.
Hence, in the multiply-imaged region, each subtraction removes all images of the
source plane clean component. In the course of modeling the lens in MG0414, we
found this freedom leads to non-physical solutions. In other words, the converged
lens model requires the source brightness distribution contain negative flux.
Using the singular isothermal sphere (SIS) plus external shear model (see Sect;ion A.1), we modeled the MG0414 system with the CLC algorithm. We optimized
the lens model parameters to fit the CLEAN image from the 15 GHz VLA observations (Katz & Hewitt 1995), shown in Figure 3-5. We followed the minimization
.
I
,
I
1
I
,
,
I
.
.
I
I.
0
o
o,
a
4) 0
O~~
0
c
-
,I
I
I
-2
-3
,~~~~..
-1
-1
RelativeR.A. (arcsec)
MAef.I:
Contou.
0.1584 JY/BEkA
():
-0.38 0.38 0.75 1.50 3.00
.00 12.00 24.00 48.01 g96.01
Figure 3-5: Final CLEAN image of MG 0414+0534 from a 15 GHz VLA observation of
MG 0414+0534 (Katz & Hewitt 1995) used in the Clean map LensClean analyses. The
contour levels are -1.5, -0.75, -0.375, 0.375, 0.75, 1.5, 3, 6, 12, 24, 48 and 96 percent
of the peak, 158.4 mJy/Beam. The beam FWHM is 0'!12 by 0"11.
59
procedure described in Section 3.2. Figure 3-6 shows the residual image using the best
fit lens model. We find
0mu,
= 215 pJy/Beam and a peak residual of 2.0 mJy/Beam.
O
0
O
:.
.
o
a
0
0
0';
7
o
-1
-1
o1
0
O .
0@
..I........1..
1
0
-1
Relative R.A. (arcsec)
Maxiwm: 1.9796E-03 JY/BEAM
Contou. (X): -60.01 -30.01 30.01 60.01
Figure 3-6: Residual Map from Best Fit SIS plus external shear model using the
Clean map LensClean method including negative clean components. The absolute
contour levels are the same as Figure 3-5.
From VLA monitoring of MG0414, Moore & Hewitt (1995) find variability in the system at the level of a few percent. Since there is a time delay between the arrival times
between each image, we expect the variability to lead to deviations from the true flux
ratios and to prevent LensClean from finding a perfect fit. LensClean constructs the
source brightness distribution and attempts to fit the observed data such that the
mean square residual is minimized. Therefore, LensClean minimizes the error in the
source plane weighted by the image magnifications.
Consider a two-image gravitational lens where the source is unresolved. If the
source is variable and the time delay is non-zero, then the mean square value in the
image is,
(x2) = [F(t)Pl]2 + [(t + At)s 212
60
(3.27)
= [ts]2+ [(t +SI)12]2
(3.28)
where yet is the source plane flux at time t and S5 is change in the source plane flux
from time t to t + At. We assume that we know the exact lens model for the system.
If we subtract a source plane component of flux Ft from the data, the mean square
residual in the image is,
C2= (S5/
2.
(3.29)
2)
However, if we subtract a component of flux Ft + SF, the mean square residual is,
2
If I1 >
2,
= (F,1)
2.
(3.30)
the smallest residual is achieved by subtracting a component equal to
the image plane flux of component 1 divided by its magnification (t).
lll <
2,
the optimal source flux is the flux at component 2 divided by
However, if
2
(t + SF).
Therefore, we see that the optimal flux for the subtracted component is determined
by the flux of the highest magnification component. The time delay between Al and
A2 is short (
1 day), so the measured A1/A2 flux ratio in the image should be very
close to the true flux ratio. Therefore, the dominant error in the modeling should be
located at component B. At 15 GHz, the total flux of B is _ 65 mJy. At a variability
level of a few percent, we would then expect peak-to-peak errors of
2 mJy/Beam
for a perfect lens model inversion. From the residual image, we find the peak errors
in the CLC inversion at components Al and not at Al or B indicating variability was
not limiting the fit.
We found a problem in the solution from examining the distribution of clean components, shown in Figure 3-7. We see that to the west of B, there is a concentration
of clean components in a region with no observed flux. The clean components are
arranged such that the large positive and negative components cancel when convolved
with the beam, resulting in no net flux in the image. We mapped these components
through the lens to find their associated components in the image and found that they
map to Al, A2, and C. This behavior is more clearly understood using the multiplicity
61
.
.
.
.
.
I.
.
.
I
1
ID
P
us
O
o
To
0
.>
Q:
_o
-1
I
1
0
-1
Relative R.A. (arcsec)
hxlmum: 0.1277 JY/BEAM
Contou
(): -25.00 -15.00 -5.00
Contou. (X): 75.00 85.00 95.00
5.00 15.00 25.00 35.00
45.00 55.00 65.00
Figure 3-7: Distribution of clean components used in the best fit SIS plus external shear model using the Clean map LensClean method including negative clean
components.
image (Figure 3-8). Each pixel in the image maps back to a single source position. If
the source is within the multiply-imaged region, then more than one image pixel will
trace back to that same location. The multiplicity image gives the number of image
pixels that map back to a given source position as a function of the image positions.
We expected that each component in MG0414 maps back to the same source, i.e. Al,
A2, B, and C are all in the four-image region. However, Figure 3-8 clearly shows B is
in the two-image region and the components west of B are in the four image region.
Somewhat surprisingly, the model has enough freedom also to map the associated
images of B to near the position of C.
We see this behavior in all the lens models used to fit MG0414. Even though the
lens models have different angular structures and radial profiles, the addition of negative components artificially enhances the fit. The key to this behavior in LensClean
is the differential magnification in the image. The components west of B are distributed such that they add no flux, however, their associated components at Al and
62
O
0I
0
a,
-1
1
0
-1
Relative R.A. (arcsec)
Figure 3-8: Multiplicity image from the best fit SIS plus external shear model using
the Clean map LensClean method including negative clean components. The curves
are boundaries between the 1, 2, and 4 image regions. Al, A2, B, and C show the
locations of the MG 0414+0534 components and + symbol shows the location of the
clean components west of B.
A2 contribute a large amount of flux. Since we are treating the image pixels as the
real photons (i.e. tracing them through the lens), negative pixels are clearly artificial
and cannot contribute to a valid image reconstruction. We, therefore, have modified
LensClean so that it subtracts only positive clean components.
This requirement
constrains the pixels to be positive, but it also prevents any compensation for oversubtracting pixels during the iterative process. We minimized the oversubtraction by
reducing the loop-gain (see Section 2.5.2) from 0.2 to 0.075.
3.5
Visibility LensClean
The Clean map LensClean algorithm forces the lens model to fit not only the gaincorrected visibilities, but the entire reconstructed visibility plane.
63
Since standard
deconvolution methods are "local" in the sense that they ignore correlations caused
by the lens galaxy, lensed images of a common source need not be consistent with
each other in the final clean map. Chen, Kochanek, & Hewitt (1995) find that for lens
model reconstructions of the Einstein ring, MG 1131+0456, differences in the CLEAN
reconstructions affect the CLC results. In particular, they examine the sensitivity
of their results to changes in the loop-gain parameter.
They found the differences
in the lens model solutions are consistent with the conservative confidence levels
adopted, further justifying the validity of these conservative error estimates. However,
variations in the loop-gain change the final rms residual by up to 50 percent. Since
the tests were limited to CLEAN images, they could not fully explore the systematic
errors, but they showed that the LensClean results are affected by the systematic
errors in the reconstructed images.
We developed an improved LensClean algorithm, named Visibility LensClean
(VLC), to avoid the systematic errors of the standard image reconstruction methods by operating directly on the complex visibility data. We implement VLC in the
NRAO Software Development Environment. The algorithm is a modification of the
Cotton-Schwab CLEAN algorithm (Schwab 1984; Cornwell & Fomalont 1989). As in
the CLC algorithm, the minor cycle cleans the image under the constraints of the lens
model. During the major cycle, the accumulated clean components from the previous
minor cycle are subtracted from the ungridded visibility data, which are then Fourier
transformed to produce the new residual dirty image. The major cycle ends when
the peak-to-peak residual does not drop by more than one percent of its value from
the previous major cycle.
Since VLC operates on the ungridded visibility data, the definition of a X2 statistic
is straightforward. Let Nvis be the number of measured visibility points. Each visibility has complex value V and noise ri. We assume that the noise shows sufficiently
little variability over the V to allow us to replace i by an average noise per visibility
point rav. If the model visibilities are Vim° del then the X2 statistic for the fit of the
64
model to the data is,
xvl=
(3.31)
2z
E
There is no ambiguity in the visibility-based clean as to the number of data being
fit and the correct form for the X2 statistic. We, therefore, determine the lens model
parameters by minimizing X2 .
To evaluate the goodness-of-fit, we need to determine the number of degrees of
freedom in the system. If we fit Ncompclean components in the source model and M
parameters in the lens model, then the number of degrees of freedom is,
Ndof = 2N,i-
3Nomp-
M.
(3.32)
Each complex visibility has two degrees of freedom and we lose three degrees of freeclom in specifying the position and flux of each clean component. Because the lens
model completely determines the relationship between image plane components and
the source plane components, we count the clean components in the source plane. The
ambiguity in Equation 3.32 is whether multiple clean components can be subtracted
at the same position. We believe Ncompis the number of independent components.
Clean components at the same location correspond to multiple sinusoids with the
same spatial frequency, and even though the clean components are chosen independently, fitting these identical functions to the measured data should not add degrees
of freedom. We, therefore, conclude that clean components subtracted at the same
location are not independent and should be treated as a single component.
We expect that the VLC results are less affected by systematic errors, and the
formal errors more closely resemble the true errors. We use the same procedure for
determining the parameter confidence levels as in CLC except we use the formal
AX2 < 4 limit for calculating the 95 percent confidence intervals. We still try to
compensate for any remaining systematic errors by rescaling the X2 such that the
minimum X2 is equal to the expected number of degrees of freedom for the system. It
is likely that this is a correct statistical interpretation, but it can only be confirmed
65
by extensive Monte Carlo simulations. A typical LensClean optimization requires
_ 10 hours on a desktop workstation.
Therefore, to obtain 95 percent confidence
levels, one would need 2 20 optimizations requiring over a week per lens model
parameterization.
3.5.1
Self-Calibration with a Model Consistent with Lensing
If the reconstruction of the image is inconsistent with the object being a gravitational
lens, then self-calibration can reinforce the errors by adjusting the antenna gains to
maximize the agreement with the incorrect reconstruction.
Such a systematic bias
in the self-calibration process will increase the estimated noise in the image, since
the converged solution is not fully consistent with the measured data. We can test
if errors are introduced by performing a self-calibration of the visibilities using the
converged VLC model of the image. The decrease in X2 gives a measure of the errors
in the original self-calibration. This approach is not fully self-consistent because the
VLC reconstruction was found using the visibility data produced by the standard selfcalibration methods. To fully correct for the systematic biases, the self-calibration and
lens modeling need to be combined so the complex gain factors and lens parameters
are optimized simultaneously.
The gain factors used in the self-calibration must be taken into account in the
counting of the degrees of freedom, so Equation 3.32 becomes,
Ndof = 2Nvis -
3Ncomp- M -
Nin
Phase only
N9 a Phase only
2 Ngain Amplitude and Phase
(3.33)
where Ngain,is the total number of gain solution intervals.
3.6
Clean map LensClean vs Visibility LensClean
To examine the differences between the CLC and VLC algorithms, we chose two
gravitational lensing systems as test cases: MG 0414+0534 and MG 1654+1346. The
66
components in MG0414 test the reconstruction of compact sources, and the diffuse
emission in MG1654 tests the reconstruction of extended low-level flux. The results
are summarized in Table 3.1 for MG0414 and Table 3.2 for MG1654.
3.6.1
MG 0414+0534
We used the data from the 15 GHz VLA observation (Katz & Hewitt 1995). The
reconstructed CLEAN image is shown in Figure 3-5. The beam full-width at halfmaximum is 0':12 by 0'11. The peak fluxes of Al, A2, B, and C are 158 mJy/Beam,
141 mJy/Beam, 56.5 mJy/Beam, and 24.2 mJy/Beam, respectively. Using empty
regions of the image, we estimated the noise at 180 mJy/Beam, and find that the
peak-to-peak error-the
is 1.39 mJy/Beam.
maximum value minus the minimum value in the region-
The data set includes 120390 visibility points with a theoretical
noise per point of 36.8 mJy for a 10 second integration period (Crane & Napier 1989).
We parameterized the lens model as a singular isothermal sphere in an external shear
field (see Section A.1),
(r) = br +
1
2
2
yr2 cos 2 ( - 0)
(3.34)
where b is the critical radius, y is the shear strength, and 0, is the shear position
angle.
When we model MG0414 using the CLC algorithm, we find strong residuals near
all the components.
Figure 3-9a shows the residual image after subtraction of the
clean components using the best fit lens parameters.
We find large residuals near
components B and C, suggesting the lens model does not have enough freedom to
fit the measured data.
However, the Al and A2 images show alternating features,
indicative of deconvolution errors. The best fit model has amul = 651 Jy/Beam and
a peak-to-peak error of 27.2 mJy/Beam.
We find a large X2 of 9393 for 315 degrees
of freedom. If we assume normally distributed errors, we find the fit is 360oafrom an
optimal fit (i.e. X2 /Ndof = 1).
We believe that some of the residual features result from the incorrect reconstruc67
(b) MG0414+0534ReridualImage
VLCat CLCBestFit Model
(a) MG0414+0534 ResidualImage
CLCBestFit Model
7
CK
8o
-1
-1
1
ls -IM
O*.
0
RelativeRA. (arceec)
1
-4J 04.1 I
0. a
-1
1
70.
0
RelativeRA (arcsec)
O* -0.0-1.0
-
-0
0.
0
(d) MG0414+0534 ResidualImage
VLCBeet Fit Model UsingSelf-CalibratedData
(c) MG0414+0534 ResidualImage
VLC BestFit Model
O0
9
0D
7
2
I
I
I,
-1
0
o
It
-- .@;,
[."Oo,.,;~9:-
21
**0o"O
~ .-o' .
0
-1
1
0
_.
C m
0:
· ·'
Rel·tive RA (arc c)o
RelativeRA (arcsec)
0 -. 1-1-417 -7.4 7. 147
O
7731n
*
JW
-:07
-1."'7_1
17311'
',
Figure 3-9: The MG+0414+0534 residual images after LensClean. The absolute
contour levels are the same as Figure 3-5. Panel (a) shows the best fit residuals after
Clean map LensClean. Panel (b) shows the residuals after Visibility LensClean using
the best fit Clean map LensClean lens model. Panel (c) shows the best fit residuals
after Visibility LensClean. Panel (d) shows the best fit residuals after Visibility
LensClean on the self-calibrated data set.
68
Model
CLC
VLCt
VLC
VLC/SC
CLC
VLCt
VLC
VLC/SC
b
xl
yt
?
Or
asec
1.179+0.001
asec
-0.480+0.003
asec
-1.296+0.002
0.092+0.003
degrees
-10.1±0.5
" ,,
,
,
,
1.181+0.001
1.181+0.001
Ocmul
-0.482+0.002
-0.490+0.001
-1.296+0.002
-1.299+0.002
0.091+0.002
0.086+0.002
-9.6+0.3
-9.6+0.3
P-P
X2
Ndof
X 2 /Ndof
pJy/Beam
mJy/Beam
651.3
842.3
795.4
788.8
27.2
25.4
23.9
23.9
9390
134000
129000
125000
315
64600
64600
61700
29.8
2.08
2.00
2.03
t We performed Visibility LensClean fixing the lens model to the best fit model found
by the Clean map LensClean algorithm.
Table 3.1: Results for the MG 0414+0534 system using the singular isothermal sphere
plus external shear model. VLC/SC indicates the results using the visibility data set
after self-calibration with the converged VLC model. The lens position is measured
relative to component B. P-P indicates the peak-to-peak value in the residual image.
.y,is measured north through east. The CLC confidence intervals are calculated using
AX 2 < 15.1 and the VLC confidence intervals are calculated using AX2 < 4. x2 is
calculated using Xc for the CLC model and X21, for the VLC models.
tions of the compact sources in the CLEAN image. The A1/A2 images are highly
magnified so small changes in the reconstruction of the B/C images are lensed into
larger features at A1/A2. CLEAN will always create these features because of its
"local" nature, raising the question whether an algorithm which "globally" fits the
data might solve the problem. To test this hypothesis, we replaced CLEAN reconstructed images with NNLS reconstructed images. The rms noise in the NNLS image
agreed with that from the CLEAN image, and the differences between the images
were less than 0.25 mJy/Beam, significantly smaller than the features seen in the
CLC residuals, but the differences are peaked about the component positions indicating changes in the point source reconstructions.
However, when we applied the
CLC method to the NNLS image, we detected no significant change in the residual
strength, pattern, or the lens model parameters. This test was not completely independent of the CLEAN algorithm because CLEAN was used to deconvolve the image
during self-calibration. As another test, we repeated the entire self-calibration using
69
the NNLS method. As before, little change was seen in the CLC results. We conclude
that the NNLS algorithm suffers from the same problems as CLEAN and introduces
systematic errors into the reconstructed images.
For the VLC lens modeling, we used the NNLS self-calibrated data set, since both
NNLS and CLEAN self-calibrated data sets perform identically in the CLC analysis.
VLC requires subtraction of each clean component from each visibility data point;
therefore, the data set was averaged from 10 second to 30 second intervals, increasing
the computational efficiency. The number of visibility points drops to 32575, each
with a theoretical noise of 21.2 mJy. First, we applied VLC using the lens model found
by the CLC method. The residuals, shown in Figure 3-9b, have significantly more
structure. Since the VLC algorithm subtracts using the dirty beam rather than the
compact clean beam, errors in the lens model are introduced throughout the residual
image via the sidelobes of the dirty beam. Consequently, we see the rms residual in
the multiply-imaged region increases to 842 1 Jy/Beam.
However, the peak-to-peak
error drops by 5 percent to 25.4 mJy/Beam, suggesting small errors in the clean map
reconstructions. The X2 is 134000; since the system has 64600 degrees of freedom,
this X2 is 194a from the expected best fit value. This value is, however, much lower
than that estimated by the CLC algorithm, although the model remains a poor fit to
the data.
Next, we optimized the lens model using VLC, and Figure 3-9c shows the residual
image. The best fit solution has amu.l = 795 /uJy/Beam and a peak-to-peak error
of 23.9 mJy/Beam, down 12 percent from the CLC best fit solution. The drop in
both the rms error and peak-to-peak error suggests systematic errors in the clean
map are affecting the CLC algorithm. We found a X2 of 129000, slightly better than
the previous model. The lens parameters were found to be consistent within the
confidence limits.
During the self-calibration stage, the deconvolution methods did not use any lensing information. As our final experiment on MG0414, we used the VLC clean component model to self-calibrate the visibility data set. This procedure tests the consistency of the gain-corrected visibility data with gravitational lensing. Because the
70
amplitudes were not adjusted in the original self-calibration, we performed a phaseonly self-calibration with a 30 second solution interval. Figure 3-9d shows the best fit
residual image after optimizing the lens model in VLC on the re-self-calibrated data.
The rms residual drops slightly to 789 mJy/Bean and the peak-to-peak error remains
unchanged. The X2lc is 125000 and the number of degrees of freedom has dropped
to 61700 through the inclusion of 2900 gain factors in the self-calibration. The lens
model parameters do vary slightly; however, the lens model does not fit significantly
better. The errors are dominated by the incorrect fit of the lens model.
The small differences between the Clean map LensClean and Visibility LensClean
solutions indicate that the CLC algorithm is affected by deconvolution errors. This
analysis also shows that the CLC algorithm underestimates the rms error, since an
incorrect lens model only causes errors near the clean components in error. The VLC
algorithm will corrupt empty regions of the residual image through the sidelobes of
the incorrect subtractions.
Therefore, the VLC algorithm allows us to more easily
identify problems with the lens model. Furthermore, we found a significant effect in
the peak-to-peak residual. The VLC/SC solution had a peak-to-peak error 12 percent
lower than the CLC solution. At the CLC best fit solution, the VLC algorithm found
the X2 was 194a from an optimal fit. After optimizing the lens model parameters
within VLC, we found a significant improvement of 14a to 180a from an optimal
fit. Even though the lens model was found to fit the system poorly, the differences
between the CLC and VLC algorithms were detectable.
3.6.2
MG 1654+1346
In contrast to MIG0414,MG1654 (Langston et al. 1989; Langston et al. 1990) provides
an example of a lensing system dominated by extended emission. The background
source is a z , = 1.74 quasar, consisting of a compact core and two radio lobes. The
southern lobe has been lensed into a ring. Kochanek (1995) has extensively modeled
this system using the CLC algorithm.
We compare our results with his models;
however, the data set used by Kochanek (1995) was unavailable so we used one that
differed slightly, in the editing. Note that these data do not show the faint bridge
71
across the center of the ring and the low-level emission stretching vertically from the
ring seen in Langston
et al. (1990).
Figure 3-10 shows the 8 GHz VLA image of
the core and lensed radio lobe with an angular resolution of 0'21 by 0''19. In empty
1652+138 8.415 GHz 07/01/89
:. o
s o.';'''::: x'~.. o......
o
0
.:
,i
Of
"
'.
'···.
.': :
;
o~~~~~
ORO
1:)
o:...
...:.'.....-...
o
o:-::o
' ei -. ;::,
[.,
2
0
-2
Relative R.A. (arcsec)
Mimum: 5.0128E-03 JY/BEAM
Contou. (X): -3.00 1.50 53.00
6.00 12.00 24.00 48.00 96.00
Figure 3-10: Final CLEAN image of MG 1654+1346 used in Clean map LensClean.
The contour levels are -3, -1.5, 1.5, 3, 6, 12, 24, 48, and 96 percent of the peak,
5.01 mJy/Beam. The beam FWHM is 0''21 by 019.
regions of the image, the peak-to-peak error in the image is 500 YJy/Beam and the
estimated noise is 59 pJy/Beam.
The data set contains 18956 visibility points, each
with a theoretical noise of 6.35 mJy/Beam (Crane & Napier 1989). The peak in the
image is 5 mJy/Beam, although most of the flux in the system is contained in the
fainter ring emission. Kochanek (1995) finds the lens in MG1654 is best fit by the
first-order approximation to the elliptical mass distribution with surface density,
= 2b2
()
2
2
(I +-
22
2+
c 2(0-Of)]
r2 + S2~,,2(8-s
(3.35)
(3.35)
where Ec = c2 DosDoL/47rGDLs is the critical surface mass density for lensing (see
Section A.5). The monopole shape is controlled by the exponent a and the core radius
72
.s. The ellipticity
and the position angle 0 govern the quadrupole shape.
Since the MG1654 system has already been well fit by Kochanek (1995), we expect
to detect more clearly small systematic errors in the image reconstructions. Our data
set had small differences from the one used by Kochanek (1995), so we found the best
fit CLC model for our data set. The residuals, shown in Figure 3-11a, have a peakto-peak error of 727 1 Jy/Beam and
Omul
=
73.8 ,uJy/Beam, leading to a X2c of 310,
significantly larger than Kochanek (1995) found. The best fit lens model found was
consistent in all parameters except for the monopole exponent ct. The residual image
shows strong arcs with alternating positive and negative flux. The largest residuals
are clustered about the emission regions of the map, and they are caused by striping
in the reconstruction of the ring (see Section 2.5.2). The two sides of the ring are
striped independently, and when the lens model in CLC overlaps these stripes, this
results in an alternating residual pattern.
We tested if the constraint that clean components be positive-definite caused
the differences in the results by using the Clean map LensClean algorithm allowing
negative clean components on our data set (CLCtt in Table 3.2). Although we do not
achieve the same goodness-of-fit, our lens model solution is consistent as Kochanek
(1995) within the conservative error bars. We also find a significantly lower residual
rms and peak-to-peak error than achieved with the positive-definite constraint.
If
we examine the clean component distribution, we find negative pixels in the image
reconstruction. Figure 3-12 shows the clean component distribution for the Clean map
LensClean analyses with and without the negative clean components. If we included
the negative clean components, we find the largest negative pixel is -45 ,Jy/Beam,
a
significant fraction of the noise level. Although the bias is significantly smaller than
that in the MG0414 analysis, we still find the negative pixels are a problem in the
MG1654 analysis.
The VLC algorithm dramatically reduces the striping problem in CLC. The best
fit VLC residual image is shown in Figure 3-11b. The ring-shaped residuals are
significantly reduced and there is no longer the alternating pattern, although some
coherent structures about the ring remain. The rms residual is omul = 61
73
Jy/Beam,
(b) MG 1654+1346ResidualImage
VLCBestFit
(a) MG1654+1346ResidualImage
CLCBest Fit
'.c.o
S
0
oo*
4o~~0~:*·
o .·
*.
QQ
IAe
a
0
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oo
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O
.§
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A
c~.
..' .' :
.O'
:*° 'c;.
i
0··
., ·
· ' ':"
'":
:xl°° ":'-"-
o.
*
2
e
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.
o.
7P
0-,
g
c
'o
e
-2
-2
0
[?~. 0:o
470 047 -.
Reatv R
0
aM C1 -47.74
2
-2
0
RelativeRA. (arcsec)
:
.:.
.
J (rcac
-3.4,14
n~.~ .ioi
(~c
2
..~:o.: ;.
0
-2
7 4.74
7.
47
(c) MG 1654+1346ResidualImage
VLC BestFit after Self-Calibration
·:'*o
^'.'..,......
;;*.o0~~~~~~~v~f
.
~ o
·
8..
O
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..- '-~"..:',.
-'0~~~~~
·;0"d
i oI 6~-o
'"'':
-'':'.--':::o"
or'o,·' :.O
, s·.~~"'
:',.,;O':'
·:~~~~~~~~'.'d
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0
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00
~
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'D
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0
~.e.
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~,
0
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0
;":~
'a~
[
f
~.
~.{:,"i...:,.
.·· ,:'..'a J.
.-..~-Z?
:: ':':.
[:P.::, C: ";..c-o
J
,~
o . .0"
.9
0
.2
0
re
-2
's.
....
['~o
[ 'o. .'.-" .- :'
.
""
'0
oi
0
2
..
oo.
.-......
'.':.'.
/0'.''0
:..
..: ...
-:.o,
-2
RelativeRA (arcsec)
-Ie4
_loee.
m
rJ/u
u
S.7T--.
1.T7
Figure 3-11: The MG 1654+1346 residual images after LensClean.
The absolute
contour levels are the same as Figure 3-10. Panel (a) shows the best fit residuals
after Clean map LensClean. Panel (b) shows the best fit residuals after Visibility
LensClean. Panel (c) shows the best fit residuals after Visibility LensClean on the
self-calibrated data set.
74
Model
a
s
b
xi
yt
1. +o.16
1.11±0.15
1.17±0.03
1.16±0.02
1.15±0.01
asec
0.5+o.oo
5
0.005±0.027
0.010±0.003
0.016±0.007
0.016±0.001
asec
1.119±0.106
1.133±0.131
1.234±0.035
1.206±0.023
1.203±0.003
asec
2.263±0.008
2.201±0.007
2.213±0.029
2.210±0.014
2.223±0.003
asec
-2.078±0.016
-2.013±0.011
-2.028±0.030
-2.032±0.015
-2.036±0.003
CLCt
0.286±0.086
degrees
11.5±1.1
CLCtt
0.283±0.066
11.6±1.9
CLC
VLC
VLC/SC
0.236±0.009
0.226±0.002
0.226±0.002
11.8±4.0
11.1±0.1
11.1±0.1
cmult
X2
Ndof
X 2 /Ndof
CLCt
CLCt t
CLC
VLC
VLC/SC
O~
pJy/Beam
P-P
mJy/Beam
CLCt
CLCtt
25
60
0.304
0.683
133
195
76
76
1.75
2.57
CLC
VLC
VLC/SC
74
61
61
0.727
0.491
0.493
310
42600
38900
76
34600
34000
4.08
1.23
1.14
t Results from Kochanek (1995) using a different data set and the CLC algorithm
allowing negative clean components.
tt Results using our data set and the CLC algorithm allowing negative clean compo-
nent.
Table 3.2: Results for the MG 1654+1346 system using the approximate elliptical
mass distribution. VLC/SC indicates the results using the visibility data set after
self-calibration with the converged VLC model. The lens position is measured relative
to the quasar core. P-P indicates the peak-to-peak value in the residual image. ,
is measured north through east. The CLC confidence intervals are calculated using
AX2 < 15.1 and the VLC confidence intervals are calculated using AX2 < 4. X2 is
calculated using X 21 for the
for the VLC models.
C1 CLC model and Xc
W~~~~~~vC
75
(a) MG 1654+1346 Clean Component Distribution
(a) MG 1654+1346 Clean Component Distribution
Clean map LensCleon without negative cleon components
·
I
Clean mop LensClean with negative clean components
~~~~~~~~~~
2
e
A
I -20
8 -2
I.0Cs
ZI
+
D
·
'
·
'
·
'''I
-2
I.
.
0
2
I
2
-2
I.
.
0
-2
Reltive RA. (arcseec)
RelativeRA (arceec)
A.m
O
e w .0 001.1 00.10
.
I
ar/jv
_/1 uT.W-4
00 5. (r -7.41 -.l
- . 5
rE-eu
Figure 3-12: The distribution of clean components using the Clean map LensClean
algorithm on the 5 GHz MG 1654+1346 data set. Panel (a) shows the distribution
using the constraint of positive definite clean components and Panel (b) shows the
distribution with negative clean components. The + symbols show the positions of
the largest negative clean components ( 15 pJy/Beam).
consistent with the noise level in the CLEAN image, and the peak-to-peak value of
491 /Jy/Beam.
The X2c is 42600. With 34600 degrees of freedom, we find X2L/Ndof
1.23, 38a from the best fit. A normal CLEAN with no lens model gives a
X2c
of 43100
with 27800 degrees of freedom, or X1/Ndof = 1.55, worse than the VLC fit. The
primary effect is that the normal CLEAN uses far more independent clean components
to fit the data. The VLC reconstruction is a more accurate representation of the image
than the normal CLEAN component model. The VLC model solution is consistent
with the CLC solution found in our analysis. However, we found that the solution is
barely consistent with that found by Kochanek (1995). While most of the parameters
are in reasonable agreement within the conservative CLC confidence intervals, the
monopole exponent a is well outside the error bars. The error in the lens position
is due primarily to the uncertainty in location of the quasar core between data sets
rather than differences in the lens models. Within our data set, the CLC and VLC
76
best fit lens positions are consistent.
As in the MG0414 system, we self-calibrated the MG1654 data using the best fit
VLC reconstruction as a model for the true image and redid a full optimization of
the lensing parameters using VLC. Again, we performed a phase-only self-calibration
since the data were not amplitude self-calibrated. The residuals are shown in Figure 311c. We found the rms residuals and peak-to-peak error did not change. However,
the Xjc, dropped to 38900 and the number of degrees of freedom has dropped to
34000, accounting for the 600 gain factors used in the self-calibration. The X2lc/Ndof
dropped to 1.14, only 19o from an optimal fit of X2/Ndof = 1, assuming normally
distributed errors. Since the lens model solution has not changed significantly from
the best fit VLC solution, we conclude that the better fit is due to the reduction of
systematic errors introduced in the original self-calibration.
The MG1654 tests show that sources dominated by extended emission are affected
by the errors in the standard deconvolution methods. First, we find VLC shows a
more dramatic effect in MG1654 than in MG0414 for two reasons. The lens model
chosen does not fit the MG0414 system, so the systematic errors were dominated by
the modeling errors. Second, the MG1654 data set contains fewer measured points
for a significantly more complex source structure.
We find that the VLC solutions
are consistent with our CLC analysis and its conservative confidence levels. The
Clean map LensClean algorithm used by Kochanek (1995) did not constrain the clean
components to be positive-definite, leading to small changes in the reconstructions
and converged lens model.
The constraints imposed by gravitational lensing allow a more ambitious interpretation of the reconstructed image. Since each of the four images is differentially
magnified, the effective resolution of these images is significantly higher than estimated by the restoring beam (Kochanek & Narayan 1992). Since we have a lens
model which better fits the system, we should be able to probe the higher spatial
frequency structure in the image by convolving the converged VLC clean component
model with a more compact beam. Figure 3-13a shows the best fit VLC reconstructed
image using the self-calibrated data set. We convolved the clean component model
77
(b) MG 1654+1346 Super-resolved Image
(a) MG 1654+1346 Reconstructed Image
VLC Best Fit after Self-Calibration
VLC Best Fit after Self-Calibration
0
o
0
0
0
@
2
2
0
0o
.-
e
-
e
%
r
I
g
10
a
0
0
®0
I2
C
to
T
-2
0
o
^
U
C
9
I
2
0
I
-2
2
RelativeRA (arcsec)
c
-l'.
is): .n u.
.-
t6.40
I a
o.
.O 57.W6
(&.Slu.
2B,
0
RelativeRA (arcsec)
-2
0a Je,603 MiLo
Figure 3-13: The reconstructed Visibility LensClean images of MG 1654+1346 using
the self-calibrated data set. The absolute contour levels are the same as Figure 3-10.
Panel (a) shows the reconstruction at normal resolution, 0'.'21by 0''19. Panel (b)
shows the super-resolved reconstruction using a 0'1 restoring beam.
of the image with a 0''1 restoring beam, an increase by a factor of two in resolution.
The super-resolved image, shown in Figure 3-13b, is free of the "striping" artifact
and appears to be reconstructed well. To test the accuracy of the super-resolution,
higher frequency VLA observations are necessary.
3.7
Discussion of LensClean
The LensClean algorithm is well suited to the problem of reconstructing extended
emission in the presence of noise and finite resolution. We found that the inclusion of
negative clean components can dramatically bias the results of LensClean. In the case
of MG0414, the extra freedom in the negative clean components caused LensClean
to find an artificial fit to the system with an error nearly as small as that expected
for an optimal fit. We found that the MG1654 system also was affected by negative
clean components, but at a much lower level. The ring systems densely sample the
78
gravitational potential and, therefore, limit the freedom with which negative clean
components can alter the fit.
We also find that the use of the clean map and beam as inputs for LensClean
introduces errors into the lens modeling results. The source of the errors is in the
incorrect reconstruction by the standard image deconvolution techniques. The primary effect is to reduce our ability to measure the goodness-of-fit of the model to the
data. The standard deconvolution constraints of positivity and compactness do not
require that the final reconstruction also be consistent with the gravitational lensing.
Modifying the LensClean algorithm to operate directly on the visibilities improves
the fit by reducing the deconvolution errors. Moreover, when a lens model is found to
fit the system reasonably well, the lensing requirements reduce the range of allowable
reconstructions and give a better fit to the data than a normal clean.
We also find that self-calibration reinforces the deconvolution errors by introducing
systematic biases into the visibility data through the gain factors. When we perform a
phase-only self-calibration on the MG1654 data using the best fit VLC reconstructed
model, we found a significant reduction in the error, implying a better fit. Although
we did not perform a fully self-consistent solution for the gain factors and lens model,
we found that a single self-calibration with the converged VLC model can reduce the
systematic errors introduced by self-calibration.
We believe that Visibility LensClean has reduced many of these systematic errors.
Consequently, we should be able to interpret the goodness-of-fit more reliably and apply less conservative confidence intervals in the lens model parameters. The reliability
of the confidence intervals depends on the sampling of the X2 surface. Sparse sampling may lead to underestimated uncertainties in the lens model parameters. Monte
Carlo simulations are required to calculate the true parameter confidence intervals;
however, these simulations are currently too computationally intensive to perform. In
agreement with Kochanek (1995), we find the approximate elliptical mass distribution is a good but not perfect fit to the MG1654 system. However, the formal errors
show the best fit lens model is no longer consistent with an exactly isothermal mass
distribution.
79
We have shown that the standard deconvolution methods used in radio astronomy
produce small systematic errors due to poor interpolation in the visibility plane and
that the errors are easily detectable in VLA images of gravitational lenses. The level
of these errors depends on the accuracy of the interpolation, and reconstructed images
from sparser arrays, such as those from VLBI or MERLIN observations, or from other
data sets containing fewer visibility points will have more serious problems. Therefore,
the rigorous analysis of radio interferometry data should be performed only on the
visibilities instead of the reconstructed images.
80
Chapter 4
Modeling the 15 GHz VLA Data
4.1
Visibility LensClean
Using the Visibility LensClean algorithm, we modeled the MG0414 15 GHz VLA data
provided by Katz & Hewitt (1995). MG0414 was observed for 3 hours yielding 32575
complex visibilities points with a theoretical noise per point of 21.2 mJy (Crane &
Napier 1989). The data were observed with the VLA in the A-configuration, resulting
a beam FWHM of ~ 01. We applied VLC to the data using a variety of lens models. Since the LensClean algorithm requires an efficient solution for the images corresponding to a given source, the lens models attempted are simple approximations of a
single galaxy. The models are characterized by a multipole expansion to quadrupole
order of either the gravitational potential or surface mass density. This approach
has been used to fit several lensing systems with satisfactory results (Kochanek &
Narayan 1992; Chen et al. 1995; Kochanek 1995).
Kochanek (1991) and Hewitt et al. (1992) used a "point solver" lens inversion
procedure (see Section 3.2) to fit the MG0414 system for the positions of the images
with a variety of simple lens models and found that the system was difficult to fit.
Components Al and A2 are near the tangential critical line and are highly magnified.
Small errors in 13/C can lead to large variations at A1/A2. Furthermore, the 15 GHz
VLA observations show Al, A2, and B to be resolved, which will add more constraints
to the LensClean inversion.
81
4.1.1
Lens Models
The lens models used sample a variety of radial (monopole) structures and angular
(quadrupole) structures. We included two lens models from Kochanek (1991). The
deflections, magnifications, and surface mass densities for the following models are
given in Appendix A.
The models specified by a gravitational potential are,
= br+
=
2
yr2cos2(0-0,)
0
b2-- (r2+ -2)a/2a
Ot
02
03
b2ln(r2 + s2)
= br + brcos 2(O-
a =0
)
1
2
+ -yr2cos2( - y)
2
Model 1 and 3 (models 3 and 2 from Kochanek (1991)) are based on the singular
isothermal sphere (SIS). Spiral galaxies show circular velocities which are constant as
a function of radius from the center of the galaxy. Therefore, the inferred dynamical
mass increases linearly with radius, indicating a three-dimensional mass density which
falls off as r- 2 . The SIS model distributes the matter in a spherically symmetric
fashion with this same radial dependence. Model 1 includes an external shear field to
break the spherical symmetry. The external shear field is the lowest order description
of a mass distribution positioned outside of the region of interest. Model 3 is an
approximation to an elliptical singular isothermal sphere. The surface mass density
is expanded into a multipole expansion and truncated after the quadrupole moment.
43 is the
gravitational potential which results from this surface mass density. Model
2 adds two parameters to vary the monopole structure.
The exponent a controls
the concentration of mass within a given radius. A smaller value of a corresponds
to a higher mass density near the lens center, where a = 0 is a Plummer potential
(Blandford & Kochanek 1987), a = 1 is an isothermal potential, and a = 2 is a
uniform sheet of matter. The core radius s is introduced to prevent the mass density
from diverging at the lens center.
82
The models specified by a surface mass density are,
=
5
(R) ( ) + externalshear
2 f x(x)dx
2 2-a (r2 +
2)
[
r2 + 2)
) cos 2(0- 0e)
2
where Ec = c2 DosDoL/47rGDLs is the critical density for lensing. Model 4 is based
on the de Vaucouleurs light distribution assuming a constant mass-to-light ratio.
I)e Vaucouleurs (1948) discovered that the light from elliptical galaxies can be described by,
I
)
= Ie-
67[(r/R)o
25
(4.1)
1]
where I, is the surface brightness at the effective radius R,.
This model uses the
same radial dependence for the monopole moment of the surface mass density. The
quadrupole moment arises from an external shear field, i.e.
kshear
=
-yr2
cos 2(0 - 0,).
Model 5 generalizes model 3 by adding a power law exponent a and core radius s
to the mass density. As in model 3, the multipole expansion of the mass density is
truncated after the quadrupole moment.
4.1.2
Results
Using VLC, we found the best fit lens model for all the lens models described in
the previous section. The results are summarized in Table 4.1. The reconstructed and
residuals images for all the best fit lens models are shown in Figures 4-1 through 4-9.
We expected that the angular structure is well constrained by the image geometry
and found the external shear models to best fit the system. For the SIS plus external
shear model (1), we found the rms residual in the multiply-imaged region aml was
795 Jy/Beam and a peak-to-peak error of 24 mJy/Beam. The rms error is dominated
by the large residuals at the image positions. The approximate elliptical SIS model (3)
is a worse fit with amul = 1.23 mJy/Beam.
We tested the dependence of the fit on the radial (monopole) structure of the
lens by fixing the exponent and core radius of model 2 to different values. Optical
83
b
arcsec
1 1.181±0.001
2 1.171±0.001
1.174±0.001
1.174±0.001
1.177±0.004
1.196±0.001
3 1.189±0.002
4 1.148±0.002
5 0.706±0.015
xl
arcsec
-0.482±0.002
-0.469±0.002
-0.479±0.002
-0.4764±0.002
-0.477±0.010
-0.474±0.001
-0.563±t0.015
-0.477±0.002
-0.482±0.010
y&
arcsec
-1.296±0.002
-1.2954±0.002
-1.296±0.003
-1.295±0.001
-1.290±0.004
-1.288±0.001
-1.311±0.011
-1.295±0.003
-1.262±0.002
a/e
0/
degrees
0.0914±0.002 -9.6±0.3
0.181±0.004 -9.7±0.3
0.144±0.002 -9.8±t0.2
0.093±0.002
-9.7±0.3
0.085±0.007
-9.7±1.0
0.085±0.007
-9.7±1.0
0.062±0.014 -9.6±2.8
0.129±0.004 -9.7±0.3
0.931±0.016 -13.1±0.3
a2 pp N
X2
2/
amut
P-P
mJy/Beam
mJy/Beam
3
0.795
0.832
0.856
0.893
0.889
0.871
1.227
23.9
26.8
27.5
26.9
27.7
25.6
28.3
4
0.830
25.6
125000
64700
1.92
5
1.490
48.2
229000
64800
3.53
1
2
Ndo.f
129000
123000
127000
130000
131000
135000
189000
64600
64700
64800
64800
64800
64600
64600
s/Re
arcsec
0.063
0.063
0.005
0.050
0.200
1.250
0.001
a
0.0
0.5
1.0
1.0
1.0
0.5
Xvlc/Ndof
2.00
1.89
1.96
2.00
2.03
2.09
2.93
Table 4.1: Visibility LensClean results on the 15 GHz VLA MG 0414+0534 data
set. The results are from Visibility LensClean on the original self-calibrated data set.
The errors are 95 percent confidence levels. The lens position is measured relative to
component B. The shear position angle is measured north through east. P-P is the
peak-to-peak value in the residual image.
observations of MG0414 do not show a nearby cluster of galaxies, so we chose radial
structures which approximate a single object, a = 0.0, 0.5, and 1.0. For the a = 0.0
and ax = 0.5 models, we set the core radius to be small, s = 0'063. To test the
dependence of the fit to the core radius, we fixed the core radius of the isothermal
model to 0'.'005,0'.'05,and 0'.'20. We found a slight preference for smaller values of a
and s, although only at the 5 percent level. Figure 4-7 shows a small amount of flux
arising near the center of the system due to the presence of the "fifth" image. Since
there is no evidence for the "fifth" image in the radio observation of MG0414 (Katz
& Hewitt 1995), we did not try larger core radii.
For the de Vaucouleurs model (4), we fixed the effective radius at 1'.25 and optimized the remaining parameters. We found this lens model performed as well as the
84
(a) Reconstructed
Image
0.~~~
(b) Residual Image
~ ~
0~~~~~~
-
.
.X
E
0A
AR-M
0
.
o
I
o
2
c)
I
o
V 0
0 T1
e
Oo
o
o
C
_1
a
-1
0
1
O
0o
0
-1
1
Relative
RA (arcseec)
Il 13
LP1aIL
u
4.
0
t
0
-1
RelativeRA (arcsec)
i a
rct
-ndP4.rl -7. i
147
Sl
s7.om
1d
a
Figure 4-1: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit singular isothermal sphere (SIS) plus external shear model
(1). The absolute contours levels are the same as Figure 3-5.
other external shear models, even though the radial structure is changed from a power
law to an exponential. The residuals have
error of 26 mJy/Beam.
Cmul
= 830 pJy/Beam and a peak-to-peak
Since we did not see a significant change in the results as a
:unction of the radial shape parameters, we did not vary the effective radius.
For the approximate elliptical mass models (model 5), we fixed a = 0.0 and
Ca =
0.5. We omitted the isothermal case (a = 1), since that model is identical to
model 3 except for the finite core radius. Since the core radius had only a little effect
.on the fit, we set s = 0'001. We could not fit the system using a = 0.0 since the
required ellipticity was much greater than unity. For the a = 0.5 model, we found a
poor fit to the system. The residual have omul = 1.49 mJy/Beam and a peak-to-peak
error of 48 mJy/Beam. The required ellipticity of 0.9 is extremely large.
These results show that none of the lens models fit the system well. The dominant
factor in the level of the fit is the angular structure of the lens. From the reconstructed
images we can see that in all cases, both components B and C are not fit well,
indicating problems with the angular shapes of the lens models. The models which
85
Image
(a) Reconstructed
(b) Residual Image
O
0
1
rs
Ie
O~
o
o{
~
e
-
0
T
I
I
A
o
9
yo
r
.2
c
0
o
0
o
o
P
. .
0
RelativeRA (arcsec)
1
8~aMr slam 11/51
0038.. 151a~
4.080.531
2.11.8
-1
1
0
-1
-1
Relative
RA (arcsec)
.lka 1.171- 2 /DM
CelN
-aM M IU
.13l
attwm
44.81I1.m
Figure 4-2: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit approximate singular isothermal ellipsoid model (3). The
absolute contours levels are the same as Figure 3-5.
Image
(b) Residual
Image
(a) Reconstructed
2
I2
-.
o
0
o.
.5
Oa
-1
-1
1
0
-1
R
Relative
RelativeRA. (arceec)
MO
l
.1.
34
1.1431W 401I .
O..
Ol. -
L
-4.44
-4
U44
1.
(arcsec)
.1
Figure 4-3: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit generalized monopole plus external shear model (2) using
a = 0.0 and s = 0'.'005. The absolute contours levels are the same as Figure 3-5.
86
(b) Residual Image
(a) Reconstructed Image
0
I
o
0
LI
A
M
I
o
o
&
a9
J:
cc
o
-1
-1
0
o0
1
-1
0
-1
Relative RA
Relative RA (arcsec)
_~.I. O1.17 s/ViM
1
. an- 1.13 l
r*s
Io
.
-41
17
Jrjff
14f-Un
- .
-47
47 LUn 13-
4
(arcsec)
7.
Figure 4-4: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit generalized monopole plus external shear model (2) using
a = 0.5 and s = 0'.005. The absolute contours levels are the same as Figure 3-5.
(b) Residual Image
(a) Reconstructed Image
0
o
0o
e
0
0 0o
0
lA
0
a.T
09
1
o
XI
9.
-1
-1
,
.
.
.
.
1
..
-1
0
1
0
Relative RA. (arcsec)
41
_...
h
. Ob
-.
"r/M
11 .4
Relative RA
-
IU 6.13 1.7
-
4
.11
It ollL-t
-1.1/
.1-
a
.
(.(~a
-1
(arcsec)
na
:Figure 4-5: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit generalized monopole plus external shear model (2) using
ca= 1.0 and s = 0'.'005. The absolute contours levels are the same as Figure 3-5.
87
(b) ResidualImage
Image
(a) Reconstructed
0
1
0
~
o
Ie
S
o
0
o
e
e
aoT
o
A:
-1
-1
:o sa
o
%
-1
0
I
0
RA. (arcsec)
Relative
Relative RA (arcsec)
Ia)..
54s4
-4.1 3107
l .7n7
W
I1J s/ 4.M IGA
-131 -131 -
-1
.34 111143 4L7 su
Figure 4-6: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit generalized monopole plus external shear model (2) using
a = 1.0 and s = 0"05. The absolute contours levels are the same as Figure 3-5.
Image
(b) Residual
Image
(a) Reconstructed
- i ..
'
Io
' I
0
0
u
a
0
~i
e
c
0
I9
e
c
>!
.
,0 o
-1
-1
0
o
1
rx
1
-1
0
0
24
m.e4 3.'Y15W m 3
a
~ON
OTII.~~ ie I11 4,1 #,,1I
Cale
-1
Relative RA. (arcoec)
Relative RA (arceec)
em..
7
a
e-n-asr/Km
O* -m
-13.7 -~
o 3.71 7.6
.11
Figure 4-7: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit generalized monopole plus external shear model (2) using
a = 1.0 and s = 0'.'20. The absolute contours levels are the same as Figure 3-5.
88
(a) Reconstructed
Image
(b) Residual Imoage
0
1
.0
0
i
o0
0O,
e
I9
O
a
0
Iv
r/
rr
-I
o°
0
- I .
0 .0.
1
., .
-1
. .
I
-1
0
RelativeRA. (arcsec)
6.l l Jr/l
Wl.. (2 0.
C11 t.? 11
Relative
RA. (arcsec)
_
14II.L
14114
l
. -
Ul-1
e-
w ol|t -IA,7-IJ.3
1CJII
./l
.
111(1
,
1 1li
Figure 4-8: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit de Vaucouleurs plus external shear model (4) using Re = 1'.'25.
The absolute contours levels are the same as Figure 3-5.
(a) Reconstructed
Image
(b) Residual
Image
e
f
I o
A
9Wo
I
y
>
I2
-1
-1
1
0
-1
1
Relative
RA (arcsec)
0C.141
.
,e77I8.1
117
eo
1-14 - U -
0
RelativeRA (arcsec)
-1
"II. ..
Figure 4-9: Panel (a) shows the reconstructed image and Panel (b) shows the residual
image from the best fit approximate elliptical mass distribution model (5) using ac=
,0.5. The absolute contours levels are the same as Figure 3-5.
89
contain internal shear contributions (models 3 and 5) do not fit the system as well as
the external shear models (1, 2, and 4).
The radial dependence of the monopole moment does not greatly impact the fit
of the lens model to the data. The Plummer potential fits slightly better than the
isothermal models. The fits favor a lower core radius, although the effect is at the
5 percent level. We can understand the weak sensitivity of the system to the radial
structure of the potential by examining the image geometry. Let rmin (rmax) be the
radial distance from the lens center to closest (farthest) image. We find the difference
between the largest and smallest radius is a small fraction of the mean radial distance
from the lens center, i.e.
(rmax
-
rmin)/2 - 0.15r, so the images only sample the
gravitational potential within a small annulus (Kochanek 1991).
We found that although the models to not fit perfectly, they do not have much
freedom to vary the parameters greatly. We found the lens positions are consistent
with the centroid of the galaxy measured by the Hubble Space Telescope (Falco 1995).
Therefore, the primary lens in MG0414 is the galaxy first detected by Schechter &
Moore (1993). The position angle of the shear is well constrained by the configuration
of the images to be -10 degrees (north through east). Even though the critical radius
and shear strength are coupled, the critical radius is well determined by the image
separations, excluding model 5 due to the poor fit and large ellipticity.
The results show that a simple monopole and quadrupole component can fit the
system at the 90 percent level (i.e. the peak residual is
10 percent of the peak
flux in the system). The primary error appears to be the position of component
C. In the SIS plus external shear model, the error in position is _ 01.
We can
imagine four sources of the error. We assumed the higher order moments in the
multipole expansion were not important in the global structure of the lens. If the
lens galaxy were deformed, the quadrupole approximation may not be valid. Second,
we have assumed that object X-the
faint object 1" west of component B-does
not
affect the lens modeling. If object X is important, the external shear will account
for some of the effect. However, the proximity of object X to the image positions
increases the strengths of the higher order moments in the multipole expansion. A
90
third source of error can arise from variability in the system. The different arrival
time of each component from the source to the observer then causes the observed
image to be inconsistent with the underlying lens. Moore & Hewitt (1995) found the
flux of MG0414 is varying at the level of a few percent. The lens inversion errors are
~ 10 percent of the peak flux which is significantly higher than would be expected
from the measured variability. The fourth possible contribution to the error is from
the global mass distribution in the universe. Seljak (1994) showed that the effects of
large-scale structure will shift the relative positions of the images by a few percent.
In the case of MG0414, the apparent shifts would be x 50 mas, slightly smaller than
our position error at C. However, structures smaller than those considered by Seljak
(1994) could significantly affect the image geometry.
4.2
Point Source Lens Inversion
Since the previous lens models were unable to fit the MG0414 system, we pursued
more complex lens parameterizations.
Although we preferred the LensClean algo-
rithm because of its proper handling of extended structures and finite resolution,
the method requires that the image positions and magnifications be determined efficiently for all possible source components. LensClean is currently not able to handle
the more general classes of models. Therefore, we developed a "point solver" inversion
technique for this system. The point solver only maps the images back to the source
plane and, thus, does not require a solution to the nonlinear equation for the images.
Since the MG0414 system consists of four compact images, we approximated those
components as point sources and required that the four images must map back to
the same point in the source plane. Therefore, both the positions and the fluxes at
the source plane must agree. Furthermore, we required that the position of the lens
must coincide with the centroid of the galaxy at optical wavelengths. Falco (1995)
determined from Hubble Space Telescope (HST) measurement the position is 0466
91
west and 1'.'27south of component B. We define our error measure as,
X2
2
2
2
(4.2)
= Xpo + Xflux + Xlens
where,
4
(
Xpos
(a))
(4.3)
(,i) is the mean position, and ap, is the uncertainty in the source plane position. In
the coordinate system where the magnification matrix is diagonal, we see that the
magnification is given by
= XAu,. Since the magnification in the two directions are
not the same, the correct error measure is [(u
2
+
(v
)2] /ro where as
is the image plane error. However, the deviation of the magnification from unity is
usually dominated by one eigenvalue. We, therefore, define the source plane error as,
caT
3
= as,/lpIl
(4.4)
where aso is the image plane error and pilt is the magnification for the ith image.
Therefore, our error measure is,
pos [
v(u _ -)2 + )A (v
_)2] /
(4.5)
which is typically larger by a factor of two than the true error measure since we
underestimated the error in the unmagnified direction(Kochanek 1991). For this
formulation, we see that the correct definition for the mean is,
=
()
E4
A
(4.6)
It12
The error measure from the magnifications (fluxes) is,
4
li= ~(4.7)
Xf
-~
(TF))2~
luxz
where Yj = fi/lupil is the source plane flux of the ith image, (F) is the square magni92
fication weighted mean flux, and a:o is the flux error in the source plane. The image
plane flux errors
f, are simply scaled by the magnification to obtain the source plane
errors,
(4.8)
aFy= afi/lpil.
To constrain the primary component in the lens model to be centered on the
position of the galaxy found by Falco (1995), we added,
(4.9)
2lens
= (Xmdel- HST)
We estimated the uncertainty in the HST position of 30 mas, approximately half the
point spread function. Since the position is measured relative to component B, the
error estimates the uncertainties in finding the centroid of both the lens galaxy and
component
B.
From the four images in MG0414, we have three relative position and three relative flux measurements from Katz & Hewitt (1995), given in Table 4.2. Since the
magnification changes rapidly near Al and A2, we only used the peak flux values and
not the integrated flux values. The HST position of the primary component gives two
more measurements. Therefore, the number of degrees of freedom is,
Ndof= 11- M
(4.10)
where M is the number of parameters required to specify the lens model.
Component
Aa
asec
As
asec
Al
A2
0.588
0.721
-1.934
-1.530
aO
Flux
asec mJy/Beam
Orf
mJy/Beam
0.01
0.01
159.12
139.98
6.84
6.02
B
-
-
-
56.335
2.42
C
-1.361
-1.635
0.01
24.153
1.43
Table 4.2: Image positions and peak fluxes for the components in MG0414. The
image positions are measured relative to component B. These data were taken from
Katz & Hewitt (1995).
93
4.2.1
Lens Models
We began with two lens models implemented in LensClean for comparison: the singular isothermal sphere in an external shear and the approximate singular isothermal
ellipsoid,
1
01
= br + 2 2r cos2(0-
02
= br+ brcos2(O-0
O)
)
Both of these models have five parameters, leaving six degrees of freedom.
We used two approaches to fit the angular structure in MG0414. The previous
models assumed that the monopole and quadrupole moments were sufficient to describe the galaxy. We tested this assumption by including higher order moments in
the multipole expansion. We chose the following models,
1
3
2
04 =
05
2
= br + -Yr2 cos2( - 0) + Ebrcos2( - 0,)
=
1
2 cos2( - ) + pr3 COS
br + -yr
3( 2
)
1
br + -yr2 cos 2(0 - 0) + r4 cos 2( 0 - 0a).
2
These models are based on the SIS plus external shear model (1). Model 3 includes
quadrupole moments from the external shear and ellipticity of the mass distribution.
Model 4 breaks the reflection symmetry assumed in the previous models and Model
5 simply includes the octupole moment. The additional terms in models 4 and 5
arise from mass distributions external to the system (i.e. outside the ring of images).
Each additional term adds two parameters-one
specify the position angle-to
to specify the amplitude and one to
the lens model, leaving four degrees of freedom in the
system.
Our second approach concentrated on the possible perturbation to the gravitational potential by object X. Schechter & Moore (1993) found the position of object
X to be 1" west of B, although they did not state an error on this determination. We
94
Lens
1
2
b
arcsec
1.187+0.017
1.192+0.053
xl
arcsec
-0.469±0.035
-0.597+0.100
y
arcsec
-1.269+0.014
-1.306+0.031
2
Xpos
7/e
0.091+0.021
0.093+0.050
0d/s
degrees
-10.3+2.9
-11.3+7.6
2
Xf lux
Xlens
1
158.6
49.8
0.2
209
2
232.5
103.9
23.3
360
Table 4.3: Best fit lens model parameters and error statistics for the singular isothermal sphere plus external shear and approximate singular isothermal ellipsoid using the
point source solver. The errors are 95 percent confidence intervals. The lens position
is measured relative to component B. These models have six degrees of freedom.
introduced a second singular isothermal sphere to several of the previous lens models
as an approximate description of object X. Since the second component adds three
parameters to specify the mass and position, we chose simple lens models for the
primary galaxy,
06
= br+b'I r-il
07
= br+ brcos2(0-0) +b'Ir'-l
0
= br + yr2cos2(0-0e)+b' Ir -irl
1
2
Model 6 consists only of two singular isothermal spheres. With this model, we explored the possibility of accounting for the ellipticity in the primary galaxy through
the effects of a perturbing spherically symmetric component. Models 6 and 7 attempt
to fit the system with the primary galaxy dominating, and using the additional SIS
component as a perturbation to match the observed images.
4.2.2
Results
We started the minimizations of models 1 and 2 at the best fit solution found by
Visibility LensClean. As expected, we found that these lens models do not fit the
system well. The results are summarized in Table 4.3.
95
Lens
3
4
5
b
xl
Yl
arcsec
arcsec
1.183±t0.025 -0.450-0.063
1.1890.004
-0.467+0.006
1.1690.016
-0.412±0.038
E/fib/b 2
1b/6b2S
a
y
arcsec
-1.302I0.075
-1.240±0.004
-1.304+0.075
0.107±0.307
0.081±0.010
0.084±0.359
degrees
6.0±19.0
-17.0±1.6
-1.3±5.7
2
Xpos
2
Xflux
2
Xlens
X2
3
4
0.030±0.091
0.004±0.002
-53.7±29.9
41.8+4.7
12.3
0.6
53.3
5.6
1.4
1.1
67
7.3
5
0.006±0.200
26.1±6.5
72.0
40.8
4.7
118
Table 4.4: Best fit lens model parameters and error statistics for lens models 3, 4,
and 5 using the point source solver. The errors are 95 percent confidence levels. The
lens position is measured relative to component B. These models have four degrees
of freedom.
Next, we examined the lens models with additional moments. The results for
these models are given in Table 4.4. We see an interesting result. The addition of
a quadrupole or octupole moment in models 3 and 5 improved the fit over model 1;
however, these models are still far from an optimal fit. We find that after adding an
"odd" component, we were able to reasonably fit the system with a x2 of 7.3, where
the system has four degrees of freedom. The dominant source of error lay in matching
the flux of component C. The cos 30 component is interesting since this is the only
lens model component which does not have elliptical symmetry. We find the cos 30
component is about a factor of 10 weaker than the quadrupole moment at the image
positions (y , 10b).
To make sure that we did not create any additional images, we used the Gravitational Image Tracing1 ray-tracing routines to map the images back to the source
plane and reconstruct the source. Then, we mapped the source forward to the image
plane. Figure 4-10 shows the reconstructed source with caustics and the image with
critical lines for model 4. Since the GrIT routines require the images to lie on pixel
centers, the images were mapped to slightly different positions in the source plane.
1
The Gravitational Image Tracing (GrIT) package was written by John Ellithorpe at the Massachusetts Institute of Technology.
96
Examination of the reconstructed image, we found no additional images produced by
this lens model. The caustics and critical lines clearly show a perturbation from the
elliptical symmetry. The tangential critical line bulges towards the northwest, in the
direction of object X.
I
OL--:
.
0tI~n
0o0
B
I
1-3
.. oo
I0
Source
,JoW
',
C.@
t
h...
::" ',
A2
...
'.,I,,,,..w';~
-t.o
Al
..1.......'..'..............
1.
-. 5
1%.
-
-
2. Is5
End
o
I-,
-
.s
i..,le
!
1.0
I.
?L..1
:ra
I
Figure 4-10: Panel (a) shows the reconstructed source and tangential caustic and and
Panel (b) shows the point images and the tangential critical for Model 4. The image
in panel (a) is a factor of four smaller in each dimension than the image in panel (b).
Due to the finite pixel size, the source components do not lie on top of one another.
For the lens models with an additional SIS component, we could fit the system,
although an ellipticity was required for the primary lens. The results are summarized
in Table 4.5. The lens model with simply two SIS components could not reproduce
the observed image positions and magnifications. The primary galaxy requires some
ellipticity apart from the effects of the second SIS component. By adding the external
shear field, we found a fit to the system with a X2 of 16, where the system has three
degrees of freedom. The approximate elliptical SIS model could not fit the system
as well. In model 8, the position of the additional SIS component is 0'.'6from the
estimated position from Schechter & Moore (1993). However, the optical detection
was near the 5o limit (Schechter & Moore 1993), so its position may be uncertain. If
object X is not at the same redshift as the primary galaxy, then its apparent position
97
will not correspond to that found through lens inversion. Furthermore, the properties
of object X change due to the different image geometry at the object X redshift. The
most significant change is that the critical radius found is the true critical radius
scaled by the ratio D2s /Dls, where D1s is the distance from the primary lens to
the source and D2S is the distance from the secondary lens to the source2 . The
size of these deviations from the solution found in the lens inversion depends on the
difference in redshift.
As with model 4, we examined the model to look for additional images created by
the lens. The critical radius of the additional component is much smaller than the
distance to any of the images, so we do not expect to find multiple imaging from this
component. Figure 4-11 shows the reconstructed source with caustics and the image
with critical lines for model 8. Again, the positions in the source plane are not perfect
due to the finite pixel size in GrIT. As expected, we found no additional images in
the system created by the lens model. The caustics and critical lines look similar to
those seen for model 4.
Since we are finding lens models which fit the system reasonably well, we are
concerned with biasing the lens model solutions from variability. In section 3.4, we
showed the peak errors due to variability will occur at component B because of its
large time delay and differential magnification with respect to Al and A2. To test
the level of corruption in the fit, we changed the flux of B and fit the system using
2
For a system with a second lens at a different redshift, the lens equation is,
Dooss = DosO-Dsi5- D
-.
B=s~Di
(
Dos
--.s
1
2
62
D s 2)
where O indicates the observer, S indicates the source, 1 indicates the primary lens, and 2 indicates
the secondary lens. We have assumed that the secondary lens does not produce multiple images.
Note that
is the angular position of the apparent image and 1 (2) depend on the angular
positions of the light ray at the primary (secondary) lens plane. We can transform this lens into a
single object at the primary lens redshift. The deflection angle of the single lens is given by,
D2S _,
= 1- D
Dls
2-
The exact form of this lens depends on the primary and secondary lens redshifts. From the transformation, we see that the secondary lens deflection is scaled by the ratio D 2SIDls.
98
Lens
b
X
Yl
6
7
8
arcsec
0.851±2.000
0.949±0.075
1.127+0.047
arcsec
-1.029+0.500
-0.504+0.044
-0.441±0.008
arcsec
-1.304±0.075
-1.407+0.025
-1.290+0.005
b'
xi
Y
6
7
8
arcsec
0.450+1.500
0.318±0.166
0.076+0.063
arcsec
0.049+0.375
-0.981±0.500
-0.454+0.275
arcsec
-0.211+0.500
-0.589±0.875
-0.199+0.656
2
Xpos
2
Xflux
Xlens
X
1560
5.3
3.5
46.2
10.3
11.3
436
22.0
1.2
2040
37.6
16.0
6
7
8
2
/7
#e/
0.103+0.011
0.088+0.020
degrees
-18.7+2.0
-13.0+3.2
Table 4.5: Best fit lens model parameters and error statistics for lens models 6, 7, and
8 using the point source solver. The errors are 95 percent confidence levels. The lens
positions are measured relative to component B. Model 6 has five degrees of freedom
and models 7 and 8 have three degrees of freedom.
the cos 30 model. Figure 4-12 shows the change in X2 as a function of the B flux.
WVesee that a change in flux of 3 mJy (5 percent) changes the X2 by g, 3. We
also find the change in the converged lens parameters are well below the 95 percent
confidence intervals listed in Table 4.4. Therefore, we do not believe the variability
level measured by Moore & Hewitt (1995) significantly affects the lens modeling. It
is interesting to note the minimum is at a flux of 2 mJy lower than the measured
flux, which indicates possible variability in the system; although the lens model is
probably not perfect.
The two models which best fit the system have ellipticities due to mass outside
the ring of images. However, the HST images of the galaxy show an ellipticity of
- 0.3 (Falco 1995). Both models also indicate mass towards the position of object X.
Angonin-Williams et al. (1994) stated that object X was most likely a galactic star,
although the Hubble Space Telescope observations show no evidence for a compact
stellar object (Falco 1995). Diffuse emission is seen around the position of object
X, supporting the interpretation of object X as a galaxy. Furthermore, the Hubble
99
TOONE.,
B
.
:oE
....
/'
1
,'
CD,
or!: .
Source
o.
~ ~ ~ ~ ~ ~~'.
0,00
"..~~~~~~~~~~~~~~~~~~~
[,.]O
.
'
.
.
A2
X
C
Al
,.
...........................
I::
2.:S
.........................
. ...... ................
V,,.a:
·m
, ' 2.',3
~.,
-I.L
. .. ..........................
.. ,),,:
1i5.:2
M.:3
mnrr
Irr a
mmmrrrrrmrr
-rmrmrrmrrrrr
1
~
~
_
1~_1
~ ~
_
.
.1~1~__~~
__
..
Figure 4-11: Panel (a) shows the reconstructed source and tangential caustic and and
Panel (b) shows the point images and the tangential critical for Model 8. The image
in panel (a) is a factor of four smaller in each dimension than the image in panel (b).
Due to the finite pixel size, the source components do not lie on top of each other.
observations do not show the lensing galaxy to be distorted significantly towards the
northwest, although rigorous analysis of the lens has yet to be done (Falco 1995). We
find that the cos 30 model fit the system better than the two SIS plus external shear
model. However, the structure of both lens models suggests a similar interpretation.
The cos 30 model places the "perturbing" mass external to the lensing galaxy, causing
the same distortions as the two component SIS model. We believe that object X does
influence the lensing geometry of the system and should be taken into account in
further modeling of the system.
100
or
--
I
.
.
.
'
'
.
'
.
.
I
I
.
I
.
.
f
.
|
.
I
.
.
t
I
I
I
I
.
10
9
I
8
7
I
52
,I
,
I
I
I
54
56
ComponentB Flux [mJy]
I II
58
Figure 4-12: Plot of X2 versus the flux of component B. We used the cos 30 lens model.
The x2 is the value after optimizing all the lens model parameters. The Eosymbol
indicates the measured flux of component B used in the lens inversion.
101
Chapter 5
5 GHz VLBI Observations
5.1
Observations
On June 7th, 1992, MG0414 was observed at 5 GHz with an array of ten telescopes:
Effelsberg, Greenbank, Haystack, Medicina, the phased-VLA, and 5 VLBA1 telescopes: Fort Davis, Kitt Peak, Los Alamos, Owens Valley, and Pie Town. The
European VLBI Network (EVN) stations observed from 12:00 UT through 16:00 UT,
when MG0414 set. The North American stations observed from 12:00 UT through
23:00 UT. At each non-VLBA station, the observation was made with the Mark III
VLBI recording system in mode B using a 28 MHz bandwidth (Rogers 1983). The
VLBA stations recorded the signals using the VLBA system in a Mark III mode B
compatible format.
Since the MG0414 components are separated by no more than 2", they were within
the primary beams of all the single dish telescopes. The phased-VLA primary beam is
the interferometric beam and not the primary beam of each individual VLA antenna.
We observed the source using the compact D-north-C array configuration, resulting
in a beam size of -
10". Therefore,
we observed all of the MG0414 components
simultaneously.
'The Very Long Baseline Array is part of the National Radio Astronomy Observatory which is
operated by Associated Universities, Inc. under co-operative agreement with the National Science
Foundation.
102
The observation was divided into blocks of approximately one hour in length, each
corresponding to four 13 minute tape passes. At the start of each block, we observed
0423-013 (J2000), a redshift z = 0.9 BL Lac object, as a bright calibrator. Previous
VLBI observations of 0423-013 revealed little structure at the 1 milliarcsecond level
(Wehrle et al. 1992). Although the flux of 0423-013 has varied dramatically, it usually
remains greater than 1 Jy, which allows us to use short 90 second scans of 0423-013
in the phase calibration and "clock finding" (see Section 5.2). The remainder of each
block of time was dedicated to observing MG0414.
Effelsberg, Greenbank, Haystack, and Medicina required short interruptions to
measure system and antenna temperatures.
Even though the VLA and the VLBA
stations measure both as the observations proceed, we introduced a short break between each block in the schedules at all the stations to keep the tapes synchronized.
Some difficulties were experienced during the observations. Before the observation,
the formatter in the Greenbank Mark III data acquisition rack was left disconnected.
The formatter provides accurate timing information to the tape recorder; therefore,
the data from Greenbank could not be correlated. The recording head in the Kitt
Peak VLBA tape recorder jammed at 19:42 UT, resulting in a loss of data on all Kitt
Peak baselines through 22:06 UT.
5.2
Correlation
The data tapes were shipped to the Haystack Observatory for correlation on the Mark
III processor. Each station provided an observing log, calibration information, and
data for the a priori correlator model (see Section 2.6.2). The geometric information,
such as the exact location of each telescope, was known through the results of previous
geodesy experiments.
The tropospheric moisture content was estimated from the
weather information in the observing log.
The Mark III processor correlates each station pair at eight residual delays (i.e.
eight delays with respect to the correlator model) from -ps
to lpis. Instrumental ef-
fects must be known to this accuracy to measure the complex response of the program
103
source. We used the instrumental offsets (known as "clock" offsets) found from other
observations nearby in time as starting values. To provide a strong unambiguous
signal at the correct offsets, we correlated a 0423-013 scan near the midpoint of the
observation. If a strong peak was not found (i.e. the "clocks" were off by more than
Alps), we varied the offsets until the peak shifted into the response window. This
procedure is known as "finding clocks." Correlating scans at the beginning and end
of the observation, we found the clock offsets were stable throughout the observation,
and these values were introduced into the correlator model as constant instrumental
offsets.
The delay beam for the longest baseline (
108 A) is 3'.'5, which results in more
than a 10 percent amplitude reduction for objects 1" from the delay center. Even
though all the components in MG0414 were observed simultaneously, the size of the
delay beam required that the observation be correlated at multiple delay centers to
prevent bandwidth smearing (see Section 2.4.1). We chose three positions at which
to correlate the MG0414 scans: At the midpoint between Al and A2, at the position
of B, and at the position of C. The delay center coordinates, shown in Table 5.1,
were obtained from high-resolution 15 GHz VLA observations of MG0414 (Katz &
Hewitt 1995). We chose to correlate at a position between Al and A2 rather than
at both Al and A2 to reduce the correlation time (approximately 3 days per delay
center). The bandwidth smearing due to the 0'.'25offset of Al and A2 from the delay
center resulted in well under a one percent amplitude reduction.
Delay Center
A1/A2
B
C
0423-013
ca
m
0 4 h 14
37.7759s
h
14"
37.7322s
04
m
0 4 h 14 37.6411P
" 15.8006s
0 4 h 23
050
050
050
-010
34'
34'
34'
20'
42.5415"
44.2690"
42.6390"
33.0627"
Table 5.1: Correlation delay center coordinates. The MG 0414+0534 positions are
taken from Katz & Hewitt (1995) and the 0423-013 position is from the VLA Calibrator Manual. All positions measured with respect to the J2000 coordinate system.
The Mark III correlator calculates the complex visibility function from the signals
104
recorded at each station. For each station pair, the correlator introduces a delay for
one station ranging from -3ps to +1ps in increments of ps. For each of these eight
delays, the two station signals are corrected by the correlator model and multiplied
together. At this point, the phase changes are small enough so the visibilities can
'be averaged over an accumulation period of 2 seconds. The integration increases the
signal-to-noise ratio in each visibility point and also significantly reduces the total size
of the data set. The correlated data are archived onto digital-audio-tapes (DATs) at
the Haystack Observatory for further processing.
5.3
Calibration
'The MG0414 and 0423-013 data sets were imported into the NRAO Astronomical
Image Processing System (AIPS) for fringe fitting and amplitude calibration. Each
data set corresponding to a single delay center was processed independently.
The
subsequent sections describe the calibration procedures. The AIPS tasks FRING and
ANCAL were used to perform the global fringe fitting and amplitude calibration.
15.3.1 Fringe Fitting
At most VLBI stations, a phase-calibration signal is injected into the signal to measure
the time-variable instrumental delays. However,at the time of the observation, the
VLBA stations had not been equipped with the phase-cal systems. Since 0423-013
provided an excellent calibration source, we performed a "manual phase-calibration"
by fringe fitting for only the delay in each frequency channel (single-band delay)
on scans of that source. In addition, differences between the Mark III and VLBA
electronics cause a 120 degree phase shift between the upper and lower sideband
signals of each frequency channel on baselines which include one Mark III station and
one VLBA station. We compensated for this phase shift by solving for each sideband
separately in the manual phase-calibration.
Since no 0423-013 scans were available in which all stations were present, we chose
to calibrate using the scans at 14:36 UT and 15:51 UT. Fort Davis was chosen as the
105
reference antenna since it was present in both scans and operating well. The delay solutions for stations common to both scans were found to be consistent. The solutions
were applied throughout the entire observation. We verified that the instrumental
delays did not change during the observations.
We fringe fitted the 0423-013 scans using a point source model for the source.
We used the entire scan length (-
90s) for the solution. To prevent large phase
errors due to weak signal detections, we required every delay-rate solution to have a
signal-to-noise ratio larger than seven on at least three baselines.
In order to fringe fit the MG0414 scans, a model of the source structure was
necessary. Since a change in the delay or rate shifts the direction at which the interferometer points, a source containing multiple compact components will have multiple
peaks in the response as a function of delay and rate. These peaks are convolved with
the delay beam and, in the case of MG0414, are close enough such that the overlapping beams cause the peaks to shift relative to the true delay and rate values. For
that reason, we divided the MG0414 data by a model of the source structure, thereby
removing phase contributions from the source structure. The correct delay and rate
were then found at the position of the response peak.
The model was derived from the same 15 GHz VLA observations of MG0414 used
to determine the delay center positions (Katz & Hewitt 1995). The model consisted of
four point sources at the positions of the components (See Table 5.2). Since the flux
ratios appear to be consistent across many radio frequencies (Katz & Hewitt 1995),
we used the 15 GHz flux ratios to set the relative fluxes of the point sources in the
model. To set the absolute flux scale of the model, we fixed the B component flux
to be equal to that measured at 5 GHz by Hewitt et al. (1992). We fringe fitted
the MG0414 data sets using a four minute solution interval. As with 0423-013, we
required a signal-to-noise of seven on at least baselines for a valid solution.
5.3.2
Amplitude Calibration
In contrast to the VLA, VLBI observations use telescopes with different properties,
so the measured amplitudes of the complex visibility must be calibrated. All stations
106
Component
Al
A2
B
C
Aa
As
Flux
Flux
arcsec
arcsec
Ratio
mJy
0.591
0.726
04 14 37.72789
-1.375
-1.933
-1.529
05 34 44.2750
-1.634
2.591
2.287
1.000
0.369
347.19
306.46
134.00
49.45
Table 5.2: Four point source model of MG 0414+0534 used for the global fringe
fitting. The positions and flux ratios are relative to B. The absolute positions and
relative fluxes were obtained from Katz & Hewitt (1995) and the absolute flux level
was determined from Hewitt et al. (1992). All positions are measured with respect
to J2000 coordinates.
provided system temperature measurements during the observation. On Effelsberg
baselines, the gains were calibrated using antenna temperature measurements. The
Haystack, Medicina, and VLBA stations relied on established gain curves as a function
of elevation, and a priori source fluxes. The VLA records both VLBI and VLA
formats simultaneously and we therefore used the VLA observation to measure the
total source fluxes during the observation. We found the flux of 3.9 Jy for 0423-013
and 864 mJy for MG0414. The VLA was calibrated using the provided ratios of
antenna temperatures to system temperatures measured during the observation.
5.4
Image Reconstruction
The editing and mapping procedures were performed entirely within the program
Difmap2 . Since Difmap can only process single frequency channel data, the data sets
were averaged over frequency, then copied from AIPS into Difmap.
5.4.1
Editing
When observing at low elevations, the atmospheric distortions become significant
and cause large phase errors. Therefore, we applied a 15 degree elevation cutoff to
2
Difmap version 1.2 was written by Martin Shepherd, and is part of the Caltech VLBI Software
Package.
107
the data. Furthermore, each baseline was edited to remove any outlier points which
indicated antenna-based errors. For example, between 19:19 UT and 19:23 UT, all
visibilities on VLA baselines had zero amplitude and were removed.
During the first third of the observing session, the amplitudes on all Haystack
baselines dropped dramatically. Figure 5-1 shows the visibility amplitude for 0423013 on some Haystack baselines; these should be approximately constant since the
source is compact. After 13:00 UT, the 0423-013 scans show amplitudes of < 0.5 Jy,
far less than the 3.9 Jy measured with the VLA. The cause of the amplitude drop was
not found and since there were not enough good scans on the calibrator to characterize
the amplitude change over all time, Haystack data were discarded.
A "u-v crossing point" is the point of intersection in the u-v plane where the tracks
from different baselines overlap. At these points, the different baselines should measure the same visibility and, therefore, can be used to check the visibility amplitudes.
Typical VLBI amplitude errors are of order 10 percent. However, we found Effelsberg
low by a factor of 9.4 and Medicina high by a factor of 2.1. Omitting these stations
from the calculation, we found corrections of at most 22 percent on the remaining
stations. Due to the large errors and the fact that the MG0414 visibility amplitudes
indicate significant structure on angular scales much larger than those probed by the
sparse transatlantic baselines, we chose to include only the VLA and VLBA stations
in the MG0414 imaging.
During image reconstruction, the flux in the model consistently under-represented
the measured flux on the Pie Town-VLA baseline, leaving large oscillating features
in the residual image. This short baseline-approximately
every other baseline in the observation-is
three times shorter than
sensitive to lower level flux emission which
is resolved out on the longer baselines. Therefore, we removed this baseline from the
mapping.
As a result of the editing, approximately 40 percent of the data were discarded.
The final u-v coverage and dirty beam for the MG0414 data set shown in Figure 5-2.
108
0420-014
Jun
5.0
4.97 GHz
7, 1992
I
I
I
I
I
VLBAFD-HAYSTACK
4.0
3.0
2.0
1.0
HAYSTACK-MEDICINA
4.0
3.0
2.0
1.0
8:8
4.0
3.0
2.0
I
HAYSTACK-ULBA_OV
1.0
x
8:8I
4.0
'
HAYSTACK-ULBAPT
I
3.0
-2.0
-J
1.0
4.0
_
HAYSTACK-VLBA.KP
3.0
2.0
1.0
9:8
4.0
HAYSTACK-VLBA_LA
3.0
2.0
1.0
.:8
4.0
3.0
2.0
I
HAYSTACK-ULA
I
I
1.0
12
16
UNIVERSAL
20
TIME
Figure 5-1: Visibility amplitude for 0423-013 (J2000) on Haystack baselines. The
amplitudes dramatically drop after 13:00 UT.
109
I0N
<
--
-Z~~
-
al
..
o
<
I
.
... ---'L
100
-, 50
0
~~~~~~~-~=·-----
a
a
l
l.-l -
>'
I~~~~~~~~~~..n]
=:
...
>E
o
0
I0
To
z -50
I0
I1`
-100
-2x10
7
10750
-107
100
50
00
-50
-50
-100
-100
Relative R.A. (milliarcsec)
U (wavelengths)
.Lum:
1.000Jr/BK
():
-24.00 -12.00 12.00 24.00 48.X00 .00
Figure 5-2: The left panel shows the u-v coverage and the right panel shows the dirty
beam for the 5 GHz (A6cm) observation of MG 0414+0534 after the data were edited.
To produce the dirty beam, the data were uniformly weighted with a u-v box size of
3 pixels. The FWHM of the central lobe is 17 mas by 8 mas. The largest sidelobe is
50 percent of the peak.
5.4.2
Mapping Procedure
Since the phases were not calibrated, a phase self-calibration was necessary. For
0423-013 scans, a point source model was used as the initial model. Unlike most
VLBI sources, the MG0414 system is not centrally peaked, so a point source is not an
adequate starting model for the source structure. Instead, we initialized the antenna
phases with the four component model used in the fringe fitting.
To reconstruct the image, we used the "difference" mapping procedure. The "difference" image is the residual map that remains after subtracting the clean component
model from the visibility data. We identified the location of flux in the difference map,
and only regions believed to contain flux were cleaned, constrained by tight clean
boxes. We subtracted only a small number of components during each clean cycle to
reduce the possibility of "freezing" incorrect clean components into the model. Af110
ter a cleaning cycle, we self-calibrated the data and subtracted the new components
from the visibilities. The residual data were then Fourier transformed to produce
the "difference" map for the next iteration.
The procedure was repeated until the
residual image no longer contained any flux. The exact procedure is outlined below
with Difmap commands capitalized:
1. Calibrate visibility phases (STARTMOD).
2. Phase Self-calibration/Clean Block:
(a) Clean (CLEAN) with tight clean boxes around regions of real flux, creating
the current clean component model.
(b) Self-calibrate (SELFCAL) the visibility data adjust phases only using the
combined established and current clean component models.
(c) Subtract the current model from visibility data, incorporate the current
model into the established model, and create new "difference" image (MAPPLOT).
(d) Repeat until peak in the "difference" map no longer decreases or the selfcalibration solutions start to diverge.
3. Perform a full amplitude and phase self-calibrate (SELFCAL) with the established
model.
4. Reduce the amplitude self-calibration solution interval by half. Repeat Selfcalibration/Clean Block if the solution interval is not below 30 minutes.
5. Perform a final amplitude and phase self-calibration using the established model
and a solution interval equal to the integration period.
The first amplitude self-calibration sets an overall scale for each station over the
entire observation. In other words, the amplitude self-calibration solution interval is
equal to the length of the observation. Because amplitude variations, such as those
due to cloud cover or antenna elevation, generally change on timescales much longer
111
than the integration period, we smoothed the gain amplitudes over time. In addition,
we ended the cleaning cycles when the amplitude self-calibration solution interval
dropped below 30 minutes.
We created the final images by convolving the established clean component model
with the clean beam and adding the residual image. The noise was estimated by
measuring the root-mean-square (rms) value in empty regions of the final image.
5.4.3
0423-013
Since 0423-013 is a BL Lac object and an optically violent variable (OVV), we needed
to check the level of variability during the observation. We divided the VLA observation of 0423-013 into two hour segments. We measured differences up to 2 percent in
the integrated flux over the duration of the observation, which is consistent with the
level of amplitude errors at the VLA. We, therefore, used the entire data set to image
0423-013. Figure 5-3 shows the final VLBI image of 0423-013 produced using the
image reconstruction procedures described above. The data were uniformly weighted
using a u-v box size of 3 pixels, yielding a resolution of 20.2 mas by 7.6 mas.
The data are consistent with a 3.6 Jy point source. The VLA observation measured
a total flux of 3.9 Jy, indicating amplitude calibration errors at the 10 percent level.
With the limited number of antennae and scans on 0423-013, amplitude errors at
each station can more significantly affect the reconstructed zero-spacing flux of the
source. We constrained the amplitude corrections to maintain the current total flux
in the source. However, we found large (
10 percent) amplitude errors, causing the
current total flux to underestimate the flux in the system.
5.4.4
MG 0414+0534
To reduce the computation and editing time, the MG0414 data were averaged into
30 second integration intervals. Time-average smearing effects were not significant,
since we included only the shorter baselines of the VLA and VLBA stations. Although
the data were correlated at three delay centers, the delay beam was large enough
112
100
50
E
.
0
-50
-100
100
50
0
-50
-100
Relative R.A. (milliarcsec)
Contoun
Bom:
3.378 J/EA
3o.rmum:
(X): -. 38 0.38 0.75 1.50 3.00 6.00 12.00 24.00 48.00 96.00
FWHI 20.17 x 7.59 (milbrr),
p.o. 21.4'
Figure 5-3: Final 5 GHz image of 0423-013 (J2000). The data have been uniformly
weighted with a u-v box size of 3 pixels. The size of the restoring beam is given in
the lower left corner.
to include correlated flux from more than one component.
Therefore, we needed
to image all four components in each data set. In order to adequately sample the
beam (2 4 pixels across the central lobe of the dirty beam) and include all the
correlated flux, a single image would have required over 20002 pixels, mostly empty.
Although Difmap does not have the ability to map multiple fields simultaneously,
Difmap does provide the ability to shift the position of the map (phase) center on the
sky. By modulating each visibility with e2
ri
(
'A
- \),
where u is the projected baseline
corresponding to the visibility point, the location of the phase center is shifted by
/'xf. Using this feature, we mapped multiple fields by shifting to the location of
each component during each cleaning cycle. This method of multiple field cleaning
is susceptible to biases caused by distant sidelobes in the dirty beam corrupting the
positions of the! peaks. However, the components were separated by large enough
distances so that this was not a significant problem.
Figure 5-4 shows the final VLBI maps of the MG0414 components. The image of
113
each component was created from the data correlated at the delay center located nearest that component. As in the 0423-013 mapping, the data were uniformly weighted
with a u-v box size of 3 pixels. Although there were slight differences in each data set
due to lost data during correlation, the resulting beam sizes were all approximately
17 mas by 8 mas. We chose a linear pixel size of 1 mas. The image parameters are
given in Table 5.3. The position (Aca, AS) is the location of the peak component in
each image relative to the peak in B.
Both Al and A2 are dominated by a strong compact component. In Al, a second
extended component lies to the east of the peak with ten percent of the flux of Al. The
total integrated flux of Al is 243 mJy. The A2 image has a more complex structure.
The peak flux is 121 mJy/Beam.
Along the NE-SW direction, the image appears
stretched with a bright component to the north and a possible fainter extension to
the south. The object directly to the south of the central peak is significantly brighter
than the noise level in the image. However, this component lies directly on a strong
sidelobe (
50 % of the central lobe; see Figure 5-2) of the peak component in A2.
The total integrated flux of A2 is 219 mJy.
The image of B shows a strong central component with extensions to the east and
west. The total integrated flux is 77 mJy with a peak flux is 38 mJy/Beam.
The
position angle of the extended emission is 116 degrees, in excellent agreement with
the elongation of B in the 15 GHz VLA observations of MG0414 (Katz 1994). The
structure in B also agrees with the structure seen the 5 GHz MERLIN observations
of MG0414 (Patnaik
1993).
The image of C shows a single faint component. Although this data set has been
correlated at the position of C, the delay beam is large enough to include correlated
flux from Al, A2, and B. Therefore, these brighter components must be reconstructed
well before C is detectable. However, the other components are smeared due to the
bandwidth and time-averaging, resulting less reliable reconstructions.
From these
effects, we believe this detection of C is questionable.
Katz & Hewitt (1995) measured 5 GHz integrated fluxes of 401, 362, 156, and
58 mJy for Al, A2, B, and C, respectively. The integrated fluxes in Table 5.3 show
114
A2
A2
B
_ ·_ ·_ _·-
_ _ _ -._ _ r
.
. . .
B
. . .I
V'
V
100
,0
g
-
''
I
....
100
i
50
0
.
50
ts
T
o
es
o
A
>
*:
9_
1W-,50
I-50
3
-100
W
,;
-100
ae
100
50
0
-50
100
-100
50
Relative RA. (milliarcec)
wl
I1.74 Jrl~M
C (.l. -3.0 -I.1.
.
7 7.L
1.3.1 1474.3
i;'
'
0.:
-50
-50
-100
4330
"
'
0
Relative RA. (milliarcsec)
Cr1k
100
.
(
1t1
L
(7I7
.
.
i
13
100
)
E!
,I9':' t "O
% Ae
*,'4'l-. t)
.2
.
0
A
' -50
t -50
: %
-100
O:
0
100
-100
% %
50
0
-50
-100
Relative RA. (milllarceec)
._
.
(3l -7.7 7o7l 404.4 4l 7
.17.13 -3 -I'l
I .u1
7-
Relative RA
.16
Il.n
1
P
- 0.37 *3
I4
I. (
(milliarcsec)
3
.
), Ip. 17.1
Figure 5-4: Final 5 GHz image of each component in MG 0414+0534. Clockwise from
the bottom left, the images are Al, A2, B, and C. The data have been uniformly
weighted with a u-v box size of 3 pixels.
The contour levels are -2, -1, 1, 2, 4, 8,
16, 32, 64, and 96 percent of the peak flux in Al (185 mJy/Beam). The size of the
restoring beam is shown in the lower left corner of each panel. The image parameters
are given in Table 5.3.
115
Component
Act
Peak
AS
arcsec arcsec mJy/Beam
Integrated
a
VLA Flux
mJy
mJy/Beam
mJy
Al
A2
B
+0.580
+0.725
-
-1.947
-1.541
-
186.2
123.7
37.7
242.5
219.2
76.7
1.4
2.3
1.6
401
362
156
C
-1.345
-1.454
9.1
9.1
1.5
58
Table 5.3: Parameters from the 5 GHz VLBI images of MG 0414+0534. The positions
(Ac,AS) are measured from the peak in the image to the peak in the B image. a is
determined in empty regions of the final images. The 5 GHz VLA integrated fluxes
are taken from Katz & Hewitt (1995).
that the VLBI images only recovered approximately half of the flux. The reconstructed 0423-013 image indicates the amplitude errors are at the 10 percent level.
The excess flux on the on the Pie Town-VLA baseline clearly indicate that the VLBI
images do not reconstruct all the flux in the system. Moreover, the 5 GHz MERLIN observations (Patnaik 1993) show more extended emission than detected in our
VLBI images. Therefore, we conclude that the VLBI observations resolved out the
low surface brightness emission.
The peak positions in Al and A2 relative to the peak in B are in good agreement
with the relative positions measured from the 15 GHz VLA observations. However,
the relative position of component in C is incorrect by larger than the 15 GHz VLA
beam. We, therefore, conclude that the detection of C is spurious.
5.5
Image Reliability
The primary goal of these data is to further constrain the gravitational lens inversions
of MG0414. Hence, we examined the reliability of the final images in three ways. We
checked for differences in the image reconstructions as a result of minor changes in
the data reduction procedure. We also examined the differences between the three
data sets corresponding to different delay centers. Second, we tested the stability of
the reconstructions to changes in the u-v sampling. Our final test was to check the
consistency of the final reconstructions with the lensing hypothesis, by mapping clean
116
components from one image through the lens to the other image locations.
5.5.1
Consistency within the Data Sets
We tested the reliability of some of the components in the images by not including
them inside clean boxes during the reconstructions. This experiment tests whether
these components result from errors in the image reconstruction.
Features due to
real flux should remain in the residual image convolved with the dirty beam. Errors
in the reconstruction should be corrected during self-calibration. We found that the
reconstructions of Al and B were consistent. All the components in A2 were present
in the residual image, even though they were not included within any clean boxes.
However, their fluxes were not as stable as those in Al and B.
We checked that the reconstructed images from all the delay center correlations
gave consistent results. Although rigorous comparison of the fluxes and structure was
not possible due to bandwidth and time-average smearing effects, the images from
each of the three data sets revealed the same general structure. These tests do not
fully explore the systematic effects since the data sets are essentially the same.
5.5.2
Time Segmentation
We tested the reliability of the reconstructions by segmenting the data sets in time
and comparing the results with that from the entire data set. The time segmentation
serves to change the u-v sampling function. Real structures in the image should not be
affected by the structure of the dirty beam. On the other hand, the smaller data set
provides fewer constraints on the image reconstructions. We found that splitting the
data in half caused significant problems in the image reconstruction procedures. The
smaller data set led to unreliable reconstructions because of the poor u-v coverage.
'The visibility data before 16:00 UT were noisier than the rest of the observation, so
we chose to map the data from 16:00 UT through 23:00 UT.
Since we do not have a reliable detection of C, we focus only on the reconstructions
of Al, A2, and B. Figure 5-5 shows the reconstructed images of A1/A2 and B. The
117
A1/A2
*
S
'
.
X
-
o
·o
-
a~~~~~~~~~~~
o
200
.
200
·a
-
.o."
°
o° ;.
E
o
0~~~~~~~
0
Of
8
a
· *o
-200
r~~~~~~~~~~~
o.C
_
-200
E
,
·
I
'
'I
,
I'
-200
0
I
200
0
Relative R
C.l
eor hjl .
1-.I
A1
LI
s
3
X
Cou
b5 IIR 13 II0tIle (.m q
-200
200
I
I
0
-200
Relative RA. (milliarcsec)
(milliarcec)
A....?.-.U
em
1t 21
41
l. SJ
W.
~ -4 141
kp.c lM IMW
6.43
Ill IU
1
11 11J 4,.11m
(dmWr).IL p.II .
Figure 5-5: Final 5 GHz images of A1/A2 and B using data from 16:00 UT to
23:00 UT. The data have been uniformly weighted with a u-v box size of 3 pixels.
The contour levels are -3, -1.5, 1.5, 3, 6, 12, 24, 48, and 96 percent of the peak in Al
(193.8 mJy/Beam). The size of the restoring beam is shown in the lower left corner
of each panel.
image parameters are shown in Table 5.4. The beam FWHM is now 16 mas by 12 mas.
As before, the segmented data were uniformly weighted with a u-v box size of 3 pixels.
We increased the linear pixel size to 2.5 mas so that the images of both Al and A2
fit within the same map. The structure in Al is similar to the structure seen before.
The larger beam size causes the component to the east to merge with the central
peak. The faint extension to the north is also seen, indicating it is probably real. The
peak flux in Al increased to 194 mJy/Beam, although the total integrated flux has
dropped by seven percent. The A2 image shows differences in its reconstruction. The
integrated flux in A2 is well below that reconstructed using the entire data set. The
higher noise level in the A1/A2 image is due to residual sidelobes caused by errors
in the reconstruction. The fainter extension to the south and the component directly
south of the peak are not present and probably not real. The other components
in A2 are all present, although with different relative strengths. The fluxes for the
118
northern component and the component to the southwest do change; however, both
components remain above the noise level. We conclude these components are probably
real.
This reconstruction of B results in a lower overall noise level than in the entire
data set. The peak is 54 mJy/Beam and the total integrated flux in B has increased
to 114 mJy. The noise level has decreased to 0.8 mJy/Beam since more of the lower
level emission has been correctly reconstructed, reducing the level of sidelobes extending to the empty regions of the image. The structure is the same as the structure
seen previously, except with more low-level emission present. Hence, we believe the
structure in VLBI image of B is real.
Component
Al
A2
B
Peak
Integrated
a
mJy/Beam
mJy
mJy/Beam
193.8
116.2
53.7
224.8
155.9
113.5
1.8t
1.8t
0.8
Table 5.4: Parameters from the 5 GHz VLBI images of MG 0414+0534 using only
the data from 16:00 UT to 23:00 UT. a is determined in empty regions of the final
images. t The noise level for Al and A2 was determined from the same image.
5.5.3
Consistency with Gravitational Lensing
We took advantage of the fact that MG0414 is a gravitational lens system to test
the fidelity of the reconstruction. Since we expect the components of MG0414 to be
distorted images of one another, we compared the reconstructions of the images by
mapping them through a lens model and comparing the results. This test does not
rigorously test the reconstructions, but rather verifies the coarse structure since we
do not have a lens model which fits the system perfectly (see Section 4.1). We chose a
generic lens that could reproduce the gross image geometry: the singular isothermal
sphere in an external shear field model. Since the MG0414 components are resolved
in the VLBI maps, the exact lens position relative to the components is not known
119
precisely. We convolve the Al clean component model from the VLBI maps to the
resolution of the 15 GHz VLA observation to find the offset of the peak in Al. If the
various components in the VLBI images have different spectral indices, the position
of the lens center will be incorrect. The lens model parameters are given in Table 5.5.
b
x
Y
arcsec
arcsec
arcsec
1.196
0.468
-1.290
7
Ol
degrees
0.081
-75.6
Table 5.5: Parameters for the singular isothermal sphere plus external shear lens
model used to ray-trace the Al clean component model. The lens position (,y) is
measured relative to B and x increases to the west. 0 is measured north through
east.
We performed the ray-tracing with the Gravitational Image Tracing (GrIT) package. Since the Al reconstruction appears reliable, we used the Al clean components
as the model of the true Al image. Mapping back to the source plane, we reconstructed the source plane brightness distribution, given in the upper right panel of
Figure 5-6. Because of the finite pixel size, the fluxes will not be preserved in the raytracing. The effective resolution of the source is increased by the magnification of the
lens. The large ellipticity of the restoring beam is caused by the strong magnification
in only the tangential direction at Al. The reconstructed source has been convolved
with the magnification-corrected clean beam. The peak is 0'.384 west and 1224 south
of component B. The unlensed source is consistent with a core-jet structure; however,
multifrequency or polarization observations are needed to distinguish a core from a
jet.
We mapped the source brightness distribution to the four image positions. The
reconstructions are given in Figure 5-6. We included Al to check for consistency in the
ray tracing and found Al to be reconstructed correctly. The A2 image is commonly
expected to be a mirror image of Al, since they are near the tangential critical line.
However, we see that the A2 component appears significantly different from Al. The
two components in Al are widely separated in A2. The relative separation between
Al and A2 is smaller by
-
50 mas indicating an error in the lens position. The
120
Reconstructed
Source
B Reconstruction
100
10
7.9
T
I
E
II~~~
.g
0
i
:
-10
-100
10
0
-10
100
0
Relative
RA (milliorcsec)
RelativeRA (milliarcuec)
~
PC -IJo
1.55
LSo
SM V
N
-100
uI 1 ooM
C Reconstruction
A1/A2 Reconstruction
II %
200
100
eII
.2
E
E
0
0
.1
8
-200
II
D
..
.
200
100
I
0
-200
100
RelativeRA (milliorcsec)
W
c.Iuaio
LA ItS M
%
-100
0
Relative
RA. (milliarceec)
LS M
W
O*1-u
-Jo
M UL 5MtOM M 4
se
Figure 5-6: The upper right panel shows the reconstructed source brightness distribution of MG0414 from the clean component model of Al. The reconstructed source
has been convolved with the lens distorted clean beam. The other panels contain the
reconstructed images of MG0414 using the reconstructed MG0414 source. The source
and images were created using the Gravitational Image Tracing (GrIT) package. The
brightest two components are indicated in each panel with the Roman numeral: I is
the brightest component and II is the second brightest component at Al.
121
A2 reconstruction does not show the additional components seen in A2 VLBI image.
The reconstruction of B does strongly resemble to the VLBI image of B, providing
good evidence that MG0414 is a gravitational lens and that the VLBI image of B is
reliable. The C component reconstruction shows two components; however, we do
not have reliable detection of C in the VLBI images for comparison
We believe the VLBI images of Al and B are reliable models of the true image,
while some features of A2 are suspect. The ray-tracing of the Al clean components
suggest that the northern component in A2 is real and is associated with the eastern component in Al.
The southern components and the lower-level emission are
questionable in A2. Even though we have used a lens model which is known to fit
the system poorly (see Section 4.1.2), the critical line structure about Al and A2 is
similar to that in cos 30 model, which was found to fit the system reasonably well.
Therefore, we believe the changes in the A2 reconstruction indicate the final VLBI
image is suspect. The reconstruction of C appears to be spurious because its position
is not in agreement with the reliable VLA positions.
5.6
Lens Inversion
Since we have a lens model which fits the system (see section 4.2), we used the most
reliable components from the VLBI reconstructions to further constrain the model.
Our lens inversion method requires that we map point sources back to the source
plane. Therefore, we chose the two brightest components from the VLBI images of
Al. From the previous section, we found that these components are mapped to the
central and northern components in A2 and the central and eastern components in
B (See Figure 5-4). The positions and fluxes for the components are given in Table 5.6. The position errors were chosen to be approximately half the beam size for
the primary components and equal to the beam size for the secondary components.
The larger position uncertainties for the B components reflect the different magnification at B. The components have been stretched and, therefore, the positions of the
components which map to those in Al and A2 are less clear. Comparing the fluxes
122
between the segmented and nonsegmented data, we estimated a flux uncertainty of
10 mJy/Beam for the primary components. The larger 15 mJy/Beam error for the
secondary components is because the secondary components are extended and more
dependent on the reliability of the reconstruction.
Component
Aa
AS
as
Flux
a
mJy/Beam
arcsec
arcsec
arcsec
mJy/Beam
-0.577
-0.594
-0.712
-0.728
-1.945
-1.939
-1.543
-1.516
11
14
11
14
185.9
39.0
122.3
28.7
10
15
10
15
B primary
-
-
-
38.6
10
B secondary
-0.048
-0.023
22
13.8
15
Al
Al
A2
A2
primary
secondary
primary
secondary
Table 5.6: Input positions and fluxes from the VLBI images. The component positions
are measured with respect to the B primary component.
We modified the error measure from Section 4.2 to include the secondary components,
2
2
2
2
2
X = Xpos + Xpos2 + Xflul + Xfl,,
2
2
+ Xlens
(5.1)
where 1 specifies the primary component and 2 specifies the secondary component.
Since the cos30 lens model (model 4 from Section 4.2.1) was found to fit the VLA
data, we focused on this parameterization.
We began by only modeling the primary components to test the consistency of the
VLBI data with the VLA data. We started the minimization at the best fit model
given in Table 4.4. We found a good fit to the system with a X2 of 0.2 and only one
degree of freedom. The best fit model parameters are given Table 5.7.
We found the converged lens model parameters were consistent with those found
using the 15 GHz VLA data. We expected that the B components may contribute
the largest error since they were clearly resolved in the VLBI images. Using both
the primary and secondary components, we found a reasonable fit to the system,
with a X2 of 16 and seven degrees of freedom. The best fit parameters are shown in
Table 5.7. The lens model parameters did vary, although the solution is consistent
123
P
P&S 1
P&S 2
P&S+C
b
xl
Yl
arcsec
1.180±0.035
1.135±0.080
1.177±0.025
1.181±0.010
arcsec
-0.473±0.030
-0.414+0.080
-0.472±0.024
-0.463±0.012
arcsec
-1.277±0.012
-1.248±0.030
-1.273±0.015
-1.258±0.008
e/3b/3b
2
?
Or
0.096±0.063
0.097±0.043
0.098±0.040
0.083±0.017
degrees
-5.1±4.1
-16.3±6.1
-4.8±3.9
-13.9±3.4
0c/p/8
degrees
P
0.003±0.013
-6.1±25.7
P&S 1
P&S 2
0.006±0.007
0.003±0.006
38.7±17.5
-6.7±34.1
P&S+C
0.004±0.004
36.1±7.4
2
Xpos
P
P&S 1
P&S 2
P&S+C
0.001
7.9
0.002
22.1
X
2
2
Xlen
0.2
2.7
0.5
2.0
0.1
5.8
0.1
1.1
2
X2
0.3
16.4
0.6
25.2
Ndof
1
7
3
9
Table 5.7: Best fit lens model parameters and error statistics for the cos 30 model
on the 5 GHz VLBI data. P means only the primary components were fit. P&S 1
includes both the primary and secondary components were used in the inversion
and P&S 2 includes the primary component and only the fluxes of the secondary
component. P&S+C is identical to P&S 1 except that the VLA C position was
added as a constraint. The errors are 95 percent confidence levels. The lens position
is measured relative to the primary component in B. The position angles are measured
north through east.
with the VLA model within its errors. We found the dominant contribution to the
error to be in the positions of the secondary component. If we excluded the positions
from the fit (see entry P&S 2 in Table 5.7), we found a good fit to the system with a
X2 of 0.6 where Ndf = 3.
We found that without the position of the C component, the lens models had more
freedom to fit the magnifications. However, the position of C no longer agreed with
the VLA observation. We inserted the C component into the modeling by requiring
that the peak position of C measured with the VLA traced back to the same source
position as the other primary components (Model P&S+C in Table 5.7). We found
a reasonable fit with a X2 per degree of freedom of 2.8. The constraints added by the
124
position of C constrained the lens model well. The final lens model parameters agree
with those found using only the VLA data set.
We reconstructed the source brightness distribution by ray-tracing the clean component model of Al back to the source plane using the GrIT package. Then, we
mapped this source to the Al, A2, B, and C images. The reconstructed source and
images for the P&S+C lens model are shown in Figure 5-7.
We see that the source is similar to that in Figure 5-6. As before, we mapped
the source to Al to test the accuracy of the ray-tracing and found no errors. The
relative separations of the primary and secondary components for the data and best
fit lens model are shown in Table 5.8. The relative separations between the primary
Component
|Idata - data
mas
[nodel_- Yodel
mas
Al
A2
B
17.7
31.6
53.6
18.2
83.1
86.3
Table 5.8: The relative separations between the primary and secondary components
for both the 5 GHz VLBI data set and best fit lens model.
and secondary components in the A2 and B images are rather large. However, we
have made the approximation that the secondary component is point-like, which the
VLBI image of B shows this approximation in incorrect. Furthermore, we are using
the clean component model from Al, which was created using a 1 milliarcsecond
grid. Since the directions of large magnifications at Al and B are in orthogonal
directions, the error at B would be approximately
SXyB
" 10 mas. However, this
error should not significantly affect the A2 reconstruction, indicating some problems
'with the lens model or possible inconsistencies in the VLBI reconstruction of A2. The
lens inversion should be performed with higher dynamic range VLBI data and within
Visibility LensClean.
The 5 GHz VLBI data show the MG0414 system is consistent with the lensing
scenario. We find the structure of VLBI images to be reasonably described by the
125
B Reconstruction
Reconstructed Source
_
. . . . .
. . . .
__
100
10
7
I
7P
I
eI
.2
T
0
.1
II
@
'',oo
o..
Ix
-10
-100
-10
0
10
20
100
Relative RA. (milliorceec)
coi
l51U.
INO IN - O MLO M
o
II
9
o
e,~mme
I-
(1 J
0
Relative RA (milllarcec)
3Le u
lure
-100
4.ae
O 4L Im
C Recontruction
A1/A2 Reconstruction
II
100
200
I
I
.9
.2
.,
IT
II
0
0
-5
A
a
II
u1
76
II I
-200
II
,
0
200
-100
Ii
l*
N MLBILO
I II.W~
. (I>t -I1
-a-o
E
W11
JlO Izoo
0
100
-200
-100
Relative RA (mlliarcsec)
Relative RA. (milliarceec)
1
M 41.O
I I1
I M N
.
S
NM
Figure 5-7: The reconstructed images for the best fit cos 30 model. The upper right
panel shows the reconstructed source brightness distribution of MG0414 from the
clean component model of Al. The reconstructed source has been convolved with
the lens distorted clean beam. The other panels contain the reconstructed images
of MG0414 using the reconstructed MG0414 source. The source and images were
created using the Gravitational Image Tracing (GrIT) package. The brightest two
components are indicated in each panel with the Roman numeral: I is the brightest
component and II is the second brightest component at Al.
126
reconstructions from the clean component model of Al. Using the point source lens
modeling procedure, we find the best fit lens model is consistent with that found
using the 15 GHz VLA data set. Although we did not find an optimum fit to the
VLBI image, the best fit model is less than 3 from an optimal fit (i.e. X2 /Ndof = 1),
assuming normally distributed errors. The two main sources of error are the reliability of the secondary components and the resolved components in B. The secondary
components are extended and fainter than the primary components, so their exact
positions and fluxes are more uncertain. The poor quality of this VLBI observation
along with the sparse sampling in the u-v plane resulted in low dynamic range images
of the MG0414 components. The VLBA will increase the dynamic range since the
array has been used over the past years and the observational problems have been
greatly reduced. Reliably reconstructing the low-level diffuse emission in MG0414 will
provide valuable constraints on the lens model magnification structure. Second, the
VLBI images clearly show the components to be extended. Therefore, errors result
from using the peak flux and position of each component because the components are
resolved at B, but mostly unresolved at Al and A2. These finite resolution effects
are the key reason for the development of LensClean. We believe applying Visibility
LensClean to the VLBI data would significantly reduce this error.
127
Chapter 6
Conclusions
6.1
Lens Inversion Results
There is little doubt remaining that MG0414 is a gravitational lens system. The ge-
ometry of the system originally suggestedthat the sourceof the gravitational potential
was a single isolated galaxy. However, using the Visibility LensClean procedure, we
find that the system is not fit well by a simple lens model. Approximating the lens
galaxy by only the first two terms in a multipole expansion, we find the observed
data are not consistent. Although this type of lens model is successful in fitting the
lens galaxy in MG 1654+1346 (see Section 3.6.2), Chen et al. (1995) find that the
MG 1131+0456 lens is also not well described by this approximation.
By adding another term to the multipole expansion, we find a lens model which
fit the system reasonably well but not perfectly:
3 cos
(ri=br+2-yr2cos2( - ) + /3r
3( - ).
(6.1)
The best fit model parameters are given in Table 6.1. The HST I-band and V-band
images show the lens galaxy is elongated with a position angle of - 45° (Falco 1995).
The emission from the galaxy is limited to approximately ~ 0.'5, yielding little information about the structure of the galaxy outside the circle defined by the images.
However, we find the system is best fit with a lens model containing a circularly sym128
Best Fit
b
arcsec
1.181±0.010
E//b/5b
Best Fit
xi
arcsec
-0.463+0.012
2
0.004+0.004
yi
arcsec
-1.258+0.008
?
0.083+0.017
0y
degrees
-13.9+3.4
O/p/
degree
36.1+7.4
Table 6.1: Parameters for lens model which best fit the MG 0414+0534 system, the
singular isothermal sphere plus external shear and cos 30 component. The errors are
95 percent confidence levels. The lens position is measured relative to the primary
component in B. The position angles are measured north through east.
metric mass density within the images. Therefore, the luminous matter distribution
is flatter (more elliptical) than the total matter distribution.
The higher order moments are due only to a mass distribution outside the ring
of images. The quadrupole moment largely determines the image geometry, which is
further perturbed by the cos 30 moment. The external shear has a position angle of
-14° , indicating the mass distribution is located at an angle of 760 or -104° . Annis &
Luppino (1993) observed the field around the MG0414 system at infrared wavelengths
and found many faint objects within 40". Objects 5 and 9 (Table 3 from Annis &
Luppino 1993) are in the correct orientation to produce the shear, although both
objects are 2 19" away. If we assume the lens redshift is small
(zL
~ 0.1), we can
obtain a lower limit on the mass of the galaxies required to produce the shear in the
primary lens galaxy,
M(r) 10 1 h7
(19')
(Z(0.1, 2 .6 4 ,1.0)) M)
(6.2)
where Ho = 75h75 km s- 1 Mpc-l and 7Z(ZL, ZS,Q) depends on the curvature of the
universe (Equation 6.6). The dependence of M(r) on the lens redshift is identical to
that in Figure 6-1. For a lens at
elliptical galaxy (
ZL =
1, the required mass becomes very large for an
1013 h'5 M® for an object at r ~ 19").
At optical wavelengths, the lens galaxy is too faint to determine if the luminous
matter distribution outside the images can cause the observed quadrupole moment.
129
However, the HST images of the lens galaxy shows the position angle of the ellipticity
within the ring of images is not consistent with the measured shear orientation.
The perturbation to the quadrupole moment required to fit the image geometry
broke the reflection symmetry in the lens, i.e.
(r) = 5 (-r).
In particular, we
needed to include the next higher order moment from an external mass distribution,
fr 3 cos 3(0 - p). At the critical radius (approximately the distance from the lens
to the images), the amplitude of the "odd" moment is a factor of 10 lower than the
quadrupole moment ( -y
7
103b). We have verified that this level of perturbation
affects only the details of the four image geometry and does not alter the number of
images. The source of the perturbation is consistent with its being object X, the faint
object 1" west of B found by Schechter & Moore (1993). The lens model which fits
next best-the
two SIS components plus external shear model-has
the second SIS
component near the position of object X. Schechter & Moore (1993) did not quote an
uncertainty on the position, though they did state that their detection was only at
the 5 level. If object X is at a significantly different redshift than the primary lens
galaxy, the apparent position of object X does not have to agree with that found by
the lens inversion. However, the position angle of the cos 30 component should not
vary significantly.
We do not believe the cos 30 perturbation arises purely within the primary lens
galaxy. Although optical images of the galaxy do not show any obvious perturbations,
a rigorous analysis has not yet been performed (Falco 1995). Optical photometry of
elliptical galaxies has found small deviations from a purely elliptical structure (Franx
et al. 1989; Bender et al. 1987). The isophotes of many ellipticals require a cos 30
moment with an amplitude of a few tenths of a percent relative to the major axis.
Franx et al. (1989) attribute these perturbations either to dust or to true deformations
in elliptical galaxies. The magnitude of the cos 30 component scales as r-3, where rd
is the characteristic radius of the surface mass density perturbation. The asymmetries
are more concentrated at larger radii (r 2 12 kpc), so the observed deformations lead
to perturbations below 0.1 percent at the critical radius. Deformations could exist
closer to the center of the primary galaxy; however, we find within the circle of images
130
the lens is best; fit without any deformations from spherical symmetry. Deep highresolution observations of MG0414 can determine the distribution of the luminous
matter in the primary lens galaxy. Moreover, if the object X is behind the primary
lens, the image of object X should be distorted by the gravitational potential of the
primary lens. These observations can place limits on the redshift of object X relative
to the primary lens.
Seljak (1994) has shown that large scale structure will perturb the relative positions of the images by a few percent. However, structures on a smaller scale may lead
to perturbations larger than he predicted. This fact is clear if we consider that object
X is significant. Its proximity and mass cause perturbations at the one percent level.
The 5 GHz VLBI observations of the components in MG0414 show the system
to be resolved at the 10 milliarcsecond level. The lens inversion of the 15 GHz VLA
observation (100 milliarcsecond resolution) only fit for the positions and magnification
of each image. However, the resolved structures provide constraints on the gradient
of the magnification matrix at each image.
Although we do not have a reliable
detection of C, we used the data from Al, A2, and B and found the VLA lens model
is consistent with the VLBI data. However, the lack of constraints from component
C caused large uncertainties in the best fit lens parameters. We find the inclusion of
the VLA position of C in the lens inversion greatly reduces the uncertainties in the
results. The inversion results from the VLBI data show the lens model magnification
gradients are somewhat consistent with the observed VLBI data. We were limited to
the point source inversion algorithm and thus approximated the resolved structures as
point sources. The poor quality of our VLBI data set also prevented a more detailed
examination of the structure within each component in MG0414. In particular, low
surface brightness emission was not detected, causing a significant reduction in the
observed flux. These resolved structures are needed to place strong constraints on
the lens models. The recent development of the Very Large Baseline Array provides
a good instrument for high-resolution and high dynamic range-observations.
To take advantage of the resolved structure in the images, the lens inversion should
be performed using the Visibility LensClean algorithm on both the VLA and VLBI
131
data sets. The additional constraints from the extended structures will lead to smaller
confidence intervals on the lens model parameters. Furthermore, we only focused on
the angular structure of the lens by fixing the radial structure to be isothermal. The
extra information used by Visibility LensClean may also be able to constrain the
radial structure of the lens galaxy.
6.2
Optical-Radio Flux Ratio Discrepancy
The optical A2/A1 flux ratio of 0.3 (Schechter & Moore 1993) is significantly smaller
than the measured radio A2/A1 flux ratio of 0.9 (Katz & Hewitt 1995). Using the
best fit lens model, we examine the magnifications at the Al and A2 components,
shown in Table 6.2. We find that the magnifications at the primary and secondary
Component
Primary
Secondary
Al
A2
29
17
50
38
A2/A1
0.6
0.8
Table 6.2: Magnifications at the positions of the primary and secondary components
predicted by the best fit cos 30 lens model.
components change rapidly due to their close distance to the critical line. However,
the A2/A1 magnification ratios at the primary and secondary components remain
relatively constant in the range 0.6
0.8. These values are significantly larger than
the measured optical flux ratio. If the quasar core is located near either the primary
or secondary component, the optical flux ratio cannot be caused by differential magnification. Multifrequency or polarization VLBI observations are required to identify
the core component.
Our best fit lens model is identical to the P = 1.0 model used by Witt, Mao, &
Schechter (1994), except for the cos 30 perturbation.
We have shown that their lens
model parameterization (model 2 in Section 4.1 with zero core radius) does not fit
the image geometry seen in radio observations. However, the perturbation is weak
132
and should not significantly impact their results, which state that the low optical flux
ratio of 0.3 is caused by microlensing.
6.3
Nature of the Lens
Using the lens model found by the lens inversion methods, we calculate some basic
properties of the galaxy and object X. The redshift of the background quasar source
is zs = 2.64 (Lawrence et al. 1994). Unfortunately, no spectral features due to the
lens have been found (Lawrence 1995). We crudely estimate the redshift of the lens
as the most probable lens redshift for our system, zL . 0.5 (Schneider et al. 1992).
In gravitational lens systems, the total mass of the lens galaxy interior to the
critical radius M(< b) is well determined by the image geometry. For the cos 30
model, we find M(< b) is dependent only on the critical radius,
M(<
b) = rECb2 .
(6.3)
'This expression includes only the mass of the primary galaxy and not the mass causing
external shear or cos 30 moments. We can express the critical mass density in terms
of solar masses per square arcsecond,
3.1 73.1DosDOL
H0(
= =x 10h
o) DLS Mearcsec2
= 3.1 x 1011hl1(zL,
where,
(LzL,)
zs a)=
: f
Gi =
Z(1
Q2(
[(1(1
(
ZS,Q)M®arcsec- 2
- GL)(1-
(6.4)
(6.5)
- Gs) [(1 -GL)(1 - Gs)
- GLGs)
(GL - Gs)
(6.6)
1 i+
Qz, and Ho = 75h75 km s-l Mpc-l. Using Equation 3.11, we find the
mass of the primary lens is given by,
M(< b) = 4.3 ± 0.1 x 1012 h- R(zL,
133
zs, Q)M®.
(6.7)
If we assume
= 1 (Einstein-de Sitter cosmology) and a lens redshift of
ZL
=
0.5, we find the mass enclosed by the critical radius of 5.4h-1 kpc is 8.5 ± 0.1 x
10 1h-'M®. This value is in the normal range for elliptical galaxies. Figure 6-1 shows
the dependence of the lens mass on the lens redshift and Q for Ho = 75 km s- 1 Mpc - 1 .
We see that for redshifts near the source redshift of z
= 2.64, a large mass is
Bx
-.
34x
2x
0.5
1
Len
1.5
Redshift
2
2.5
[zJ
Figure 6-1: Mass of the primary lensing galaxy as a function of lens redshift. We
assume a Friedmann cosmology with A = 0 and Ho = 75 km s-l Mpc - 1. We plot
three curves corresponding to = 0.2, 1.0, and 1.5.
required to achieve the observed image separations. At redshifts larger than unity,
the curvature of the universe becomes significant.
We can estimate the mass of object X using the two SIS plus external shear lens
model. Although this model did not fit as well as the cos 30 model, it gave comparable
results. We found a critical radius for object X of 0.076 ± 0.063 arcseconds. We make
the simplifying assumption that object X is at the same redshift as the primary galaxy,
and find that the mass within the critical radius is,
M(< b) = 5.9 ± 9.6 x 109 h7R1(zL, zs, Q)Mo.
(6.8)
The mass has the same dependence on the lens redshift as the mass of the primary
134
galaxy shown in Figure 6-1. At a redshift of Q = 1 and zL = 0.5, the mass within
0.4h-1 kpc is 1L.5+ 2.4 x 109 h' 1 M o . This mass is consistent with that of a small
companion to the primary lens, although one must remember that there is no direct
evidence that these objects are at the same redshift.
The inferred velocity dispersion of dark matter in an isothermal halo is relatively
insensitive to the exact cosmological model adopted and depends only on the critical
radius of the lens,
(6.9)
4c2(Ds)
=
m
186
ODD
6b\
9
km s--.
(6.10)
Since the velocity dispersion depends only the ratio of distances, we see that
ODM
does
riot depend on H,. The critical radius for the primary lens galaxy is 1'.'181+ 0'.'010
(Table 6.1), so the inferred dark matter velocity dispersion is given by,
cDM =202
Figure 6-2 shows
0
(l+ZL)
DM
1
]GLGs
[Gs --
-
GL kms
.
(6.11)
as a function of redshift for several cosmologies and we see
that the galaxy velocity dispersion is weakly dependent on the cosmological model.
However, the redshift of the lens is important.
258
We find a velocity dispersion of
1 km/s for zL = 0.5, with a range from 220 km/s (ZL ~ 0.1) to 1400 km/s
(ZL ~ 2.5). Typically, an elliptical galaxy has a velocity dispersion of , 400 km/s,
which limits the lens redshift to zL , 1.4.
The Faber-Jackson (1976) relation connects the total luminosity of an elliptical
galaxy with its central velocity dispersion.
dispersion of dark matter
7
DM
Gott (1977) showed that the velocity
is related to the luminous velocity dispersion
0
L by
cDM = (3/2)1/2crL. However, Kochanek (1994) examined the dispersion profiles of
37 elliptical galaxies and found this relationship is true for the average of luminous
velocity dispersion over the entire galaxy, but, near the galaxy center, the luminous velocity dispersion is a good estimator of aDM. Kochanek (1994) developed a
135
1500
1000
a
0
>
S
11
0.5
1
1.5
Lens Redshift[zJ
2
2.5
Figure 6-2: Velocity dispersion of the lensing galaxy as a function of lens redshift and
Q. We plotted three values of Q: 0.2, 1.0, and 1.5.
luminosity-UDM relationship analogous to the Faber-Jackson relation:
L =L.*
where
CDM* =
(6.12)
225 ± 10 km/s and y = 4.1 + 0.9 (Kochanek 1994). In the visual band,
Mv* = -2.51og,
L + constant - -21.5 (Milhalas & Binney 1981). Therefore, we
can estimate the apparent magnitude' of the lens galaxy using,
mv = Mv. + 5log (DoL(1+ L))
5og
(DM
(6.13)
'The apparent magnitude m is measure of the flux from an object on a logarithmic scale and
given by m = -2.5 logo
10 F + c where F is the observed flux. Note that the definition is such that
a larger apparent magnitude corresponds to a fainter object. The absolute magnitude M is defined
as the apparent magnitude of an object at a distance of 10 parsecs. Therefore, the apparent and
absolute magnitudes are related by,
m-M = 5lo
lO
( 0 pc
where the left-hand side of the equation is known as the "distance modulus" and the luminosity
distance dL is related to the angular-diameter distance dA by dL = (1 + z)- 2 dA.
136
Figure 6-3 shows the dependence of the apparent magnitude as a function of redshift
and cosmology. For the redshift range from zL
=
0.1 to ZL = 2.5, we find a peak
magnitude of - 20. Angonin-Willaime et al. (1994) have measured 24.4 ± 0.5 for
the magnitude in the V-band of the lens galaxy; this result is plotted in Figure 6-3.
We applied an evolution correction (Tinsley 1972) and the K-correction (Coleman
et al. 1980) for elliptical galaxies to extrapolate the measured apparent magnitude
to a given lens redshift. We only plot the measured magnitude up to z = 2.0, since
the correction have been calculated only in this range. The corrections are made
assuming elliptical galaxies have spectra which are similar to the nuclei of M31 and
M81. These corrections are in agreement with Oke & Sandage (1968) and Pence
(1976) up to a redshift of " 1.3. However, at larger redshifts, the nuclei of M31 and
M81 may be not approximate an elliptical galaxy and the corrections may be invalid.
For a lens redshift below z - 1, we find a discrepancy of
1 magnitude,
supporting
the theory that dust is present in the lens galaxy (Lawrence et al. 1994).
14
16
i
18
2c11
20
-
EI
MI
22
24
0.5
1
1.5
LensRedshift [zJ
2
2.5
Figure 6-3: Apparent magnitude of the lensing galaxy as a function of lens redshift
and
using the Faber-Jackson relation which assumes the lens galaxy is a normal
elliptical galaxy. We have assumed Ho = 75 km s-l Mpc - 1 and plotted curves three
values of Q: 0.2, 1.0, and 1.5.
If the difference in the radio and optical flux ratios are due to microlensing (Witt
137
et al. 1994), the galaxy may not be very dusty at the image locations. If we assume
the entire galaxy is then not very dusty, we find the lens should be at a redshift
greater than unity. At these redshifts, the inferred mass and velocity dispersion of
the galaxy are 2 1.5 x 1012 h,
hM o
6.4
and 2 330 km/s, respectively.
Prospects for Determining Ho
An exciting application of gravitational lens system is the determination of Hubble's
constant. Moore & Hewitt (1995) find the radio fluxes are variable at the level of a
few percent and are pursuing the time delay measurements in this system.
To solve for Ho as a function of the measured time delay, we invert equation 3.20
to obtain,
Ho = 2(1+zL) Z(zL, zs, )
( Atl2
)
=At2 -al
-[l
()-(2)]}
(6.1
x{ [I_ _
)
(2-
2
_ _L
)]
-
[ (0) -
-2,
[-2
( 2)]}
(6.14)
-
where R-(ZL, ZS, Q) is defined in Equation 6.6. We know that the images must map
to the same source and, therefore, the lens equation gives,
-2
=-(01
(6.16)
-02)
Using this relationship, we transform Equation 6.15 to,
2 At12 )
(ZLXZS,
){
-2)*(
2
(6.17)
(Gorenstein et al. 1988b). This formulation requires the images 1 and 2 to map back
to the same source. Therefore, more observational constraints are imposed in the
calculation of H, reducing the uncertainties in the lens model from correlations in
the lens parameters.
138
In Figure 6-4, we plot Ho for a variety of lens model in an Einstein-de Sitter
cosmology (Q = 1, A = 0). Since the time delays in MG0414 have not been measured,
---///
- //
| *-*-| /1/'
'//
.....
'
500
---- SlSext+co38[VLA]
/ / // i
SlSext+cou3B[VLB]
-----
400
/ i/ I
/
/ i
// / /
SISext+SIS
.--
*
EIISIS+SIS
'
---- SlSext
only
EIISIS only
a
N
////
-300
~
*
~~~~
I, /
, ~ I i/
,.//,s
I 200
I
I!
/1'/ /
///
I,
/.,.,
//
100
.
, ,,,)"'/../;:;/:':~",
* //''7 I 11,...-
'
-~.1
0.5
1
1.5
LensRedshift[z
1,..
1
1
1
2
1
1
2.5
Figure 6-4: Hubble constant for different lens models. We have assumed a time delay
of 2 weeks between Al and B. We have assume = 1 and A = 0.
we assumed the A1-B time delay is two weeks; different delays simply scale the curve
vertically in the sense that a larger delay corresponds to a smaller Hubble constant.
At a lens redshift of zL = 0.5 and for Q = 1, the best fit cos 30 lens model predicts a
Hubble's constant of,
Ho =60±8
(2wB k
km s - Mpc - 1
(6.18)
where the quoted uncertainty is the 95 percent confidence level from propagating the
lens model uncertainties through Equation 6.17. The different models show similar
clependences on the lens redshift, although significantly different values of Ho at a
given lens redshift. The cos30 model and the two SIS plus external shear models
(SISext+SIS in Figure 6-4) are the models which fit the system the best. The other
models perform significantly worse in the lens inversion. We see that the differences
between the best fit model using the VLA data and the VLBI data are larger than
139
the uncertainty estimates, indicating that an accurate lens inversion is crucial to a
reliable determination of Ho.
We plot Ho predicted by the best fit lens model in several different cosmological
models, including those with a non-zero cosmological constant A. We vary Q from
0.2 to 1.5 in a cosmology with A = 0. We also plot three models with no curvature,
but with different values of Q and A. The cosmological models with curvature (A = 0
500
--
0=0.2 A= 0.0
-0= 1.0 A=0.0
O= 1.5 A=0.0
- -.-
n= 0.2 A= 0.8
--.
o 0.5 A- 0.5
---
400
--
/
0.8 A- 0.2
//
// /
E 300
200
100
//
,,,' /
/
1
* 7-~~~sf'-
I1
/
__
u
0.5
1
1.5
Lena Redshift
2
2.5
[zL
Figure 6-5: Hubble constant for different cosmologies and best fit lens model. We
have assumed a time delay of 2 weeks between Al and B.
and
= 0.2 and 1.5) are significantly different from the
= 1 cosmology. The
flat cosmologies with differing amounts of A show differences within the lens model
uncertainties. If Ho is found independently, this system may also be used to place
limits on curvature of the universe.
6.5 Future Prospects
The point source lens inversion methods have shown that the cos 30 models fit the
15 GHz VLA data. However, the analysis should be implemented with LensClean
to reduce the uncertainties in the lens model parameters.
140
The largest gain in the
lens modeling will be achieved by applying the Visibility LensClean algorithm to the
VLBI data. While we have shown that the lens model does also fit the magnification
gradients at the image positions, the errors were large due to the resolved structures
in the VLBI images. VLBA observations with higher dynamic range should detect
the low-level extended emission and provide more information about the magnification structure at each component. In addition, performing the lens inversion within
Visibility LensClean should provide stringent constraints on the lens model and significantly reduce the uncertainties and range of possible lens models. Therefore, one
can explore the finer details in the lens, such as the radial structure, and address the
questions of whether a lens redshift lens model is sufficient to describe the system and
whether the more terms should be included in the multipole expansion of the lens.
The best fit lens model indicate a perturbation in the gravitational potential in
the direction of object X. Object X could be a small companion to the primary
lens or simply another galaxy near the line-of-sight. Deep optical observations may
determine if object X is truly extragalactic. If object X is behind the primary galaxy,
deep images should detect distortions in the shape of the galaxy due to gravitational
lensing by the primary galaxy.
The astrophysics results to be extracted from the MG0414 system rely on knowing
the redshift of the lens. Lawrence et al. (1994) found the lens in MG0414 to be rich in
dust. If the lens contains large amounts of neutral hydrogen, the redshift of the lens
should be detectable through absorption at the A21 cm HI line. When the redshift of
the lens is known and better constrained models are obtained, this system will be an
excellent candidate for determining Hubble's constant and possibly Q.
141
Appendix A
Lens Models
This section describes the deflection angles, magnifications, and surface mass densities
for the lens models used in this thesis. The coordinates are relative to the lens center.
The rectangular coordinates have their usual definitions: (x, y) = (r cos 0, r sin 0).
A.1
Singular Isothermal Sphere (SIS) plus External Shear
Observations of the rotation curves in spiral galaxies show that to radii much greater
than the optical emission extends, the mass increases linearly with radius (i.e. the two
dimensional mass density falls off as r - 1 ). The singular isothermal sphere (SIS) has a
mass distribution which obeys this radial dependence, assuming spherical symmetry.
The quadrupole moment in this lens model arises from an external shear field. We
parameterize this model by,
1
, (r) = br + -yr 2 cos 2 (0-0)
2
(A.1)
where b is the critical radius of the lens. This model approximates the gravitational
influence from an external mass distribution by the lowest order effect, the external
shear field, which causes a quadrupole moment parameterized by the shear strength
yand the shear position angle 0. The surface mass density of the lens (not including
142
the external mass distribution) is given by,
F(
(A.2)
=FjC
2 r
where Ec = c2 DosDoL/47rGDLs is the critical surface mass density for lensing. To
simplify the computation, we define the cosine and sine shear coefficients,
y = y cos 20,
(A.3)
y, = y sin 20v
so the potential can be written as,
2-Y
2)Y+y.
0(r=br
+(x
1
(A.4)
2
The deflection produced in the x and y directions are,
a = br + X7c + y es
(A.5)
yC + x%
ay = byrI'
and the magnification is,
1
-
by2
r3 - y
.3 -7YC
P
-1
_ bzy
-
=
(A.6)
1-b =z- + C
.A.2 Generalized Monopole plus External Shear
This lens model generalizes the SIS plus external shear model by adding two parameters which control the radial dependence of the monopole structure:
{ b2ln(r2 +
2
aO
s2)
+
yr2cos 2 ( -
) .
(A.7)
a =O
The monopole exponent a determines the concentration of mass within a given radius,
in the sense that a lower value of a corresponds to a more compact mass distribution.
143
The exponent can vary from a = 0 to a = 2, where a = 0 is a Plummer distribution
(Binney & Tremaine 1987), a = 1 is an isothermal distribution, and a = 2 is a sheet
of uniform mass density. Using Equation 3.10, the surface mass density is given by,
(r) = - b2-a(r2 + S2)a/2- 2(s2 + ar 2)
2
(A.8)
The deflection produced by this model is,
a, = b2-a(r
2
+ S2)/ 2-1 +
+
y
r
a, = b2-a (r2 + S2)a/2-1 _ yc + xY
(A.)y
(A.9)
where -ycand y, are given by Equation A.3. The magnification of an image is given
by,
1 - A(r) [r2 + s2 +x 2(a - 2)] - c
A(r)xy(a - 2) - ,
1 - A(r) [r + s2 + y2(a - 2)] +
A(r)xy(a - 2) - Y,
(A.10)
where,
A(r) = b2-a(r2 + S2)a/2- 2
(A.11)
The effect of the core radius is to prevent the mass density from diverging at the lens
center. The magnification of the "odd" image (see section 3.1) is strongly dependent
on the size of the core radius. Since the "odd" image is located near the lens center,
we can approximate its magnification by,
/S\ 4-2ce
/odd
(-b
which, for a < 2, decreases to zero as the core radius tends towards zero.
144
(A.12)
A.3
Approximate Singular Isothermal Ellipsoid
The elliptical mass distribution results in a quadrupole moment different from the
SIS plus external shear lens model. We parameterize the elliptical mass distribution
by,
b
EC
(=
m
r(1 -
-2
cos 2( -
))
(A.13)
where m is the ellipticity of the mass distribution. We expand the mass density in a
multipole expansion and only include the monopole and quadrupole moments,
(r) =
-(1
+ m cos 2(9 - 0,)).
(A.14)
The potential formed by this mass distribution is,
(
= br + br cos 2 ( -
We see that the ellipticity of the potential
bution:
,) .
(A.15)
is only one third that of the mass distri-
= 'cm. We define the quadrupole shear coefficients,
ec =
bcos 20
E, =
b sin
2
(A.16)
(A.16)
0,
so the potential can be written as,
(r = br +
x2
_ y2
rr
2xy
+ r
(A.17)
The deflection produced by this lens model is,
=
Y
+
(3x2 + y2 )Ec +
=L(3y2 + x
X22))EC+
(A.8)
aro =_y(3Y2
by
+ , +3,O ~
O
145
S(A.18)
and the magnification is,
1
-
+ 3y2 (PCC
+ PSE)
bxy
_ 3xy(PCEC +
-Pss-r3
,3 Pc'c
r3
-_ (Pcc+
2
1- bX +
rPcc
Ps)
+S
+ Pes,)
(A
)
where,
pc =
Ps
A.4
=
=-s-
cos20
=
sin 20.
(A.20)
r2xr
De Vaucouleurs Mass Distribution plus Ex-
ternal Shear
De Vaucouleurs (1948) found the light distributions of elliptical galaxies are described
by,
I(() = Iee-767[0' 25-1]
(A.21)
where ~ = r/R, is the radius in units of the effective radius R,. The effective radius
is defined such that half the total luminosity is contained within a circle of radius
R,. I, is the surface brightness at the effective radius Re. This lens model assumes
a constant mass-to-light ratio and, therefore, has a mass distribution with the same
radial dependence as I(6). To add ellipticity to the lens, an external shear field is
included. The deflection produced by this lens is,
ax = b'
F(~) + X7C+ Y7s
(A.22)
a = b F() - yy + xy
where b is the critical radius, 7cand yS are given by Equation A.3, and F(6) is,
F(6) =
xI(x)dx
folxI(x)dx
146
0 <F < 2
-
-
(A.23)
(Maoz & Rix 1993). The magnification of an image is,
1 - bRe [
2
F(l ) +
bReXY[2F(5)+ (K(()] - S
'YC
-K()]
bRex [2F(S)+ fK()] - S
1 - bR,
2-Y2F()
-1
+ 2 K()] +
(A.24)
where,
( )
K() =
A.5
(A.25)
Approximate Elliptical Mass Distribution
The elliptical mass distribution is given by,
(
=
- /2 1
-0,)) +s2]
Cb2- [r2(1 + cos2(
2V
I
I
r~J\
(A.26)
We expand E (r-) in a multipole expansion to quadrupole order,
() =
b- (r2 + s2)/
2)Ot/2-1
,·
r2
1[i
[
(2
r2 + S2
) ccos 2(0 -
)]
(A.27)
We define the cosine and sine shear coefficients,
Yc =
cos 20
y, = e sin 20
so the surface mass density can be written as,
(F) =- 2
b2-(r2
+2)a/2-1 [1- (r2 + s2)
(2
-) {(x
(A.28)
+ 2x}](A.29)
(A.29)
The deflection produced by this model is,
a = b2-
cr =
[
a ,
b2- [ya-
+ Yap.) +
4
QP(ycp - Yspc)]
+ &S/IP(YcPc
+ Ysp)
- 2xbQP(^Ycps- YsPc)]
IY-(-YPC
i~r
'IOP ++Y-'Y'P~i~r
IQP r]
147
(A.30)
where,
=x2 _y2
Pc
PS
r2
=
cos 20
2xy
=
sin 20
r
(A.31)
and,
(r2 + S2)/2 _-s
aoMP
Or
1 ar2(r
OQP
Or
r(r
+ S2)a/2 - 2 2 [(r2
2
=
(A.32)
+ 2)a/2
- s]
(A.33)
a(a + 2'
r (r 2
- (r
22 (Q-OP.
S2)ac/21
2"
(A.34)
r
Using Equation 3.15, we see that the magnification matrix can be written as,
p
(
1 - az
(A.35)
1-Oy.VO
where,
a2¢
IOXX
-
Ox2
= b2 - { 2 [(r2 + 2 )c/2-1
X2c
'/C
-PC
1 OMP
r
(7r
-
y2
(
+
y 2 1 aOMP
r2 r
r
--Pcg +2g2] +
}
'Y [ -psg - pcpsgl2Pcs2]
0 y2
+
2 )a/21
OMP +
r2
d
(A.36)
2 1
aMP
r *2 r
+
Yc [y2 Pcg + pga - 2p92] +
ayyo~
'Ys
=
z
XO
=
b 2 -o
[2
Psg + pcpsg + 2cPs2]
{Y [(r2 + S
2
)a/ 2 -1
}
(A.37)
_ 2 OMP] +
Y [
Pcg-
%Ys[
Psg + p 2g + 2pg2]
cpsgi - 2pcpsg2 +
}
(A.38)
(A.39)
148
and,
=
91=
g
2
- 2r2(r + s2)/2_2
(A.40)
(r2 + 2)a/2 -1
(A.41)
1 (r2
=
2)a/2-
+
1
2
A.6
r2QP
r
3Qp
32
r
(A.42)
SIS plus External Shear and Cosine 30 Com-
ponent
This lens model contains the SIS plus external shear model and includes a cosine 30
component,
1
(r')= br + yr2 cos2 ( - 0y)+ r3 cos3 ( -0p) .
2
(A.43)
The cosine 30 component is the next higher moment caused by a mass distribution
external to the system and does not add to the surface mass density of the primary
lens. Hence, the mass density for this model is also given by Equation A.2. As with
the SIS plus external shear model, we use the definitions for the cosine and sine shear
coefficients (Equation
A.3) and define two coefficients for the cos 30 component,
p = / cos 30
p =
(A.44)
sin 30,3.
The potential can now be written as,
1
1
(r = br + 2PcY +
3
+ 3yc(
3yS
c
,) -
_ y2
=
r 2 cos20
s(Yyc - xO.s)
where,
Pt
=
~. =
2
sin2xy
2.(A.46)
=
r2
2 0.
= r2=sin
2y
149
(A.45)
The deflection produced by this lens model is,
a = b + yC+ y%9 + 3 + 3
aY=
(A.47)
- y + x7 + 3Co - 3,,,
and the magnification is,
1=
A.7
-c-
by
-b--
6x
- 6yb, -
6x
-Yas
- 6OC+6y,|
-Q(A.48)
+ 6y8
1 +
+
6+
-1
6yy&
SIS plus External Shear and Octupole
This lens model contains the SIS plus external shear model and includes the octupole
moment,
12
4
(r3 = br + )r2 cos2 ( - 0,) + r cos4 ( - 0).
2
(A.49)
Since the quadrupole and octupole terms are due to an external mass distribution,
the mass density is given by Equation A.2. As with the SIS plus external model, we
use the definitions for the cosine and sine shear coefficients (Equation A.3) and define
two coefficients for the octupole moment,
J =
cos40
J =
sin 40s.
(A.50)
The potential can now be written as,
1
1
_~~SC
q (r = br + PcY + Pss + &(3c&S+ 2Pss,) - P
(A.51)
where pc and , are defined by Equation A.46. The deflection produced by this lens
model is,
ax
b = x+ + yy, + 4(x3c + y+S) + 4,(x,
- yS)
ay=Y - YC+ xy +4,(xS, - y) -4,(xsc + y6,)
r
150
(A.52)
-=_
-
and the magnification is,
1
-
- 12/12,
- xy
r3 -as - 12P,S + 12,S
151
+
-1
(A.53)
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